Springboard Unit 1 - Equations, Inequalities And Functions 6l2i3g

Unit 1

Planning the Unit

I

n this unit, students model real-world situations by using one- and two-variable equations. They study inverse functions, composite functions, and piecewisedefined functions, perform operations on functions, and solve systems of equations and inequalities.

Vocabulary Development The key for this unit can be found on the Unit Opener page. These are divided into Academic Vocabulary and Math . Academic Vocabulary includes that have additional meaning outside of math. These are listed separately to help students transition from their current understanding of a term to its meaning as a mathematics term. To help students learn new vocabulary: Have students discuss meaning and use graphic organizers to record their understanding of new words. Remind students to place their graphic organizers in their math notebooks and revisit their notes as their understanding of vocabulary grows. As needed, pronounce new words and place pronunciation guides and definitions on the class Word Wall.

Embedded Assessments

© 2015 College Board. All rights reserved.

Embedded Assessments allow students to do the following: Demonstrate their understanding of new concepts. Integrate previous and new knowledge by solving real-world problems presented in new settings. They also provide formative information to help you adjust instruction to meet your students’ learning needs. Prior to beginning instruction, have students unpack the first Embedded Assessment in the unit to identify the skills and knowledge necessary for successful completion of that assessment. Help students create a visual display of the unpacked assessment and post it in your class. As students learn new knowledge and skills, remind them that they will be expected to apply that knowledge to the assessment. After students complete each Embedded Assessment, turn to the next one in the unit and repeat the process of unpacking that assessment with students.

AP / College Readiness Unit 1 continues to prepare students for Advanced Placement courses by: Modeling real-world situations using one- and two-variable equations. Increasing student ability to work with a wide variety of functions.

Unpacking the Embedded Assessments The following are the key skills and knowledge students will need to know for each assessment.

Embedded Assessment 1 Equations, Inequalities, and Systems, Gaming Systems Systems of equations Systems of inequalities Absolute value equations

Embedded Assessment 2 Piecewise-Defined, Composite, and Inverse Functions, Currency Conversion Piecewise-defined functions Composition of functions Inverse functions

Unit 1 • Equations, Inequalities, Functions

1a

Planning the Unit

continued

Suggested Pacing The following table provides suggestions for pacing using a 45-minute class period. Space is left for you to write your own pacing guidelines based on your experiences in using the materials. 45-Minute Period Unit Overview/Getting Ready

1

Activity 1

3

Activity 2

2

Activity 3

4

Embedded Assessment 1

1

Activity 4

3

Activity 5

3

Activity 6

2


1

Total 45-Minute Periods

20

Your Comments on Pacing

Additional Resources

Unit Practice (additional problems for each activity) Getting Ready Practice (additional lessons and practice problems for the prerequisite skills) Mini-Lessons (instructional for concepts related to lesson content)

1b SpringBoard® Mathematics Algebra 2


Additional resources that you may find helpful for your instruction include the following, which may be found in the Teacher Resources at SpringBoard Digital.

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Unit Overview

Equations, Inequalities, Functions ESSENTIAL QUESTIONS

Unit Overview

In this unit, you will model real-world situations by using one- and two-variable linear equations. You will extend your knowledge of linear relationships through the study of inverse functions, composite functions, piecewise-defined functions, operations on functions, and systems of linear equations and inequalities.

Key

As you study this unit, add these and other to your math notebook. Include in your notes your prior knowledge of each word, as well as your experiences in using the word in different mathematical examples. If needed, ask for help in pronouncing new words and add information on pronunciation to your math notebook. It is important that you learn new and use them correctly in your class discussions and in your problem solutions.



Academic Vocabulary • interpret • compare • contrast

Math • • • • • • • • • • • •

absolute value equation absolute value inequality constraints consistent inconsistent independent dependent ordered triple Gaussian elimination matrix dimensions of a matrix square matrix

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• feasible • confirm • prove • • • • • • • • • • • •

As students encounter new in this unit, help them to choose an appropriate graphic organizer for their word study. As they complete a graphic organizer, have them place it in their math notebooks and revisit as needed as they gain additional knowledge about each word or concept. Read the essential questions with students and ask them to share possible answers. As students complete the unit, revisit the essential questions to help them adjust their initial answers as needed.

How are composite and inverse functions useful in problem solving?

Unpacking Embedded Assessments

EMBEDDED ASSESSMENTS This unit has two embedded assessments, following Activities 3 and 6. They will give you an opportunity to demonstrate your understanding of equations, inequalities, and functions. Embedded Assessment 1:

multiplicative identity matrix multiplicative inverse matrix matrix equation coefficient matrix variable matrix constant matrix piecewise-defined function step function parent function composition composite function inverse function

Key

Essential Questions

How are linear equations and systems of equations and inequalities used to model and solve real-world problems?

Equations, Inequalities, and Systems

Ask students to read the unit overview and mark the text to identify key phrases that indicate what they will learn in this unit.

p. 55

Embedded Assessment 2:

Prior to beginning the first activity in this unit, turn to Embedded Assessment 1 and have students unpack the assessment by identifying the skills and knowledge they will need to complete the assessment successfully. Guide students through a close reading of the assessment, and use a graphic organizer or other means to capture their identification of the skills and knowledge. Repeat the process for each Embedded Assessment in the unit.

Piecewise-Defined, Composite, and Inverse Functions p. 99

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Developing Math Language As this unit progresses, help students make the transition from general words they may already know (the Academic Vocabulary) to the meanings of those words in mathematics. You may want students to work in pairs or small groups to facilitate discussion and to build confidence and fluency as they internalize new language. Ask students to discuss new academic and mathematics as they are introduced, identifying meaning as well as pronunciation and common usage. Remind students to use their math notebooks to record their understanding of new and concepts.

As needed, pronounce new clearly and monitor students’ use of words in their discussions to ensure that they are using correctly. Encourage students to practice fluency with new words as they gain greater understanding of mathematical and other .

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UNIT 1

Getting Ready

UNIT 1

Getting Ready

Use some or all of these exercises for formative evaluation of students’ readiness for Unit 1 topics.

Write your answers on notebook paper. Show your work.

Prerequisite Skills • Evaluating functions (Item 1) HSF-IF.A.2 • Finding slope and intercepts (Item 2) HSF-IF.B.4 • Graphing linear equations (Item 3) HSF-IF.C.7a • Writing linear equations (Items 4–5) HSF-IF.B.4 • Finding additive and multiplicative inverses (Item 6) 7.NS.A.1b • Solving linear and literal equations (Items 7, 9, 10) HSA-REI.B.3, HSA-REI.D.10 • Understanding absolute value (Item 8) 6.NS.C.7 • Finding domain and range (Item 11) HSF-IF.B.5 • Identifying lines of symmetry (Item 12) 4.G.A.3

1. Given f(x) = x2 − 4x + 5, find each value. a. f(2) b. f(−6) 2. Find the slope and y-intercept. a. y = 3x − 4 b. 4x − 5y = 15 3. Graph each equation. a. 2x + 3y = 12 b. x = 7 4. Write an equation for each line. a. line with slope 3 and y-intercept −2 b. line ing through (2, 5) and (−4, 1)

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output

−1

c.

−3

−5

y 4

y

2

8

–6 –4 –2

6

2

–2

4

6

2

4

x

–4

4 2 2

4

6

8

10

d.

x

y 4 2

–4 –6

–8 –6 –4 –2

–8

–2

–10

–4

7. Solve 3(x + 2) + 4 = 5x + 7. 8. What is the absolute value of 2 and of −2? Explain your response.

8

8

x

–6

6. Using the whole number 5, define the additive inverse and the multiplicative inverse.

y

6

–8

12. How many lines of symmetry exist in the figure shown in Item 11c?

9. Solve the equation for x. 3x + y =2 z

x=7

2x + 3y = 12 4 2 –10 –8 –6 –4 –2 –2

2

4

6

8

10

x

–4 –6 –8 –10

4. a. y = 3x − 2 b. 2x − 3y = −11 5. 3x + 4y = 24 6. Sample answer: The additive inverse of 5 is −5 because 5 + (−5) = 0 and (−5) + 5 = 0. The multiplicative inverse of 5 is 1 because 5 5 1 = 1 and 1 5 = 1. 5 5 7. x = 3 2 8. 2; Absolute value is the distance from 0 on a number line, so it cannot be a negative number. The absolute value of both 2 and −2 is 2. 2z − y 9. x = 3 10. C. Sample explanation: When −1 is substituted for x and −2 is substituted for y in the equation, you get 6(−1) − 5(−2) = −6 + 10 = 4. So, (−1, −2) is a solution.

()

2

()

2

SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions

11. a. domain and range: all real numbers b. domain: 3, 7, 11; range: −1, −3, −5 c. domain: −5 ≤ x ≤ 5; range: −1 ≤ y ≤ 1 d. domain and range: all real numbers 12. There are two lines of symmetry, the x- and y-axes.


Getting Ready Practice For students who may need additional instruction on one or more of the prerequisite skills for this unit, Getting Ready practice pages are available in the Teacher Resources at SpringBoard Digital. These practice pages include worked-out examples as well as multiple opportunities for students to apply concepts learned.


–2


–10 –8 –6 –4 –2

1. a. 1 b. 65 2. a. slope: 3, y-intercept: −4 b. slope: 4 , y-intercept: −3 5 3. a–b.

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11. Find the domain and range of each relation. a. y = 2x + 1 b. input 3 7 11

5. Write the equation of the line below.

Answer Key

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10. Which point is a solution to the equation 6x − 5y = 4? Justify your choice. A. (1, 2) B. (1, −2) C. (−1, −2) D. (−1, 2)

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ACTIVITY

Creating Equations

ACTIVITY 1

Directed

One to Two Lesson 1-1 One-Variable Equations Learning Targets:

• Create an equation in one variable from a real-world context. • Solve an equation in one variable.

Activity Standards Focus In Activity 1, students write and solve linear equations in one variable, including multistep equations and equations with variables on both sides. They also write equations in two variables and show solutions to those equations on a coordinate plane. Finally, they write, solve, and graph absolute value equations and inequalities. Throughout this activity, emphasize the importance of performing the same operation on both sides of an equation or inequality in an effort to keep the equation or inequality balanced.

My Notes

SUGGESTED LEARNING STRATEGIES: Shared Reading, Activating Prior Knowledge, Create Representations, Identify a Subtask, ThinkPair-Share, Close Reading

A new water park called Sapphire Island is about to have its official grand opening. The staff is putting up signs to provide information to customers before the park opens to the general public. As you read the following scenario, mark the text to identify key information and parts of sentences that help you make meaning from the text. The Penguin, one of the park’s tube rides, has two water slides that share a single line of riders. The table presents information about the number of riders and tubes that can use each slide.

Lesson 1-1 PLAN

Penguin Water Slides Slide Number

Tube Size

Tube Release Time

1

2 riders

every 0.75 min

2

4 riders

every 1.25 min



Jaabir places a sign in the waiting line for the Penguin. When a rider reaches the sign, there will be approximately 100 people in front of him or her waiting for either slide. The sign states, “From this point, your wait time is minutes.” Jaabir needs to determine the number of approximately minutes to write on the sign. Work with a partner or with your group on Items 1–7. 1. Let the variable r represent the number of riders taking slide 1. Write an algebraic expression for the number of tubes this many riders will need, assuming each tube is full.

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Pacing: 1 class period Chunking the Lesson #1–2 #3–5 #6–7 #8 #9 #10–11 Check Your Understanding Lesson Practice

MATH TIP

TEACH

An algebraic expression includes at least one variable. It may also include numbers and operations, such as addition, subtraction, multiplication, and division. It does not include an equal sign.

Ask students to translate each phrase to an equation.

r 2

Bell-Ringer Activity

1. Six more than twice a number c is 24. [6 + 2c = 24] 2. One-third of a number y is 45.  1 y = 45   3  3. Seven less than the product of a number and 10 is 50. [10n − 7 = 50] Discuss with students the methods they used to translate the sentences into equations.

2. Next, write an expression for the time in minutes it will take r riders to go down slide 1.

( 2r )0.75

Developing Math Language 3. Assuming that r riders take slide 1 and that there are 100 riders in all, write an expression for the number of riders who will take slide 2. 100 − r

Activity 1 • Creating Equations

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Common Core State Standards for Activity 1 HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions. HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

This lesson refers to both expressions and equations. An expression can be made up of numbers, variables, constants, arithmetic operation symbols, and grouping symbols. If an expression is algebraic, then it must contain at least one variable. Equal signs are never a part of an expression. An equation is a mathematical sentence that contains an equal sign, showing that two expressions are equivalent to each other. 1–2 Activating Prior Knowledge, Visualization, Create Representations If students are struggling with Item 2, give them a hint to multiply the number of tubes needed for r riders by the time needed for each tube to go down the slide. This expression will combine the result from Item 1 with the information that is found in the table.


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ACTIVITY 1 Continued

Lesson 1-1 One-Variable Equations

ACTIVITY 1 Show English language learners various pictures of water slides. Explain that riders climb up stairs or take an elevator to the top of the slide and then slide down the slide into a pool of water below. Ask students to share if there is any type of ride or activity similar to this in their native countries.

My Notes

100 − r 4

5. Write an expression for the time in minutes needed for the riders taking slide 2 to go down the slide.

3–5 Activating Prior Knowledge, Simplify the Problem If students are struggling with Item 3, then share this simpler but similar problem: Suppose Jaabir is aware that of the 100 riders, 46 plan on riding slide 1. How many of those riders are planning on riding slide 2? How did you find this? [by using subtraction] Now the only difference is that Jaabir does not know the exact number of riders for slide 1; however, the variable r represents slide 1 riders. Use the same method you used to find slide 2 riders—subtraction. The only difference is you do not know the exact number of each.

It may be beneficial for some students to review the properties of equality using numeric examples, rather than only the algebraic definitions. You can use the following examples to demonstrate the properties numerically: • Addition Property of Equality: Start with 8 = 8; add 4 to both sides. 8 + 4 = 8 + 4, or 12 = 12✓

(1004− r )1.25

6. Since Jaabir wants to know how long it takes for 100 riders to complete the ride when both slides are in use, the total time for the riders taking slide 1 should equal the total time for the riders taking slide 2. Write an equation that sets your expression from Item 2 (the time for the slide 1 riders) equal to your expression from Item 5 (the time for the slide 2 riders).

( 2r )0.75 = (1004− r )1.25

MATH TIP

7. Reason abstractly and quantitatively. Solve your equation from Item 6. Describe each step to justify your solution.

These properties of real numbers can help you solve equations. Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a − c = b − c. Multiplication Property of Equality If a = b, then ca = cb.

Sample work:

( 2r )0.75 = (1004− r )1.25 0.375r = (100 − r)0.3125

Simplify each side.

0.375r = 31.25 − 0.3125r

Distributive Property

0.6875r = 31.25 r ≈ 45.5

Add 0.3125r to each side. Divide each side by 0.6875.

Division Property of Equality If a = b and c ≠ 0, then a = b . c c Distributive Property a(b + c) = ab + ac

• Subtraction Property of Equality: Start with −4 = − 4; subtract 2 from both sides. −4 − 2 = −4 − 2, or −6 = − 6✓ • Multiplication Property of Equality: Start with 16 = 16; multiply both sides by −3. 16(−3) = 16(−3), or −48 = −48✓ • Division Property of Equality: Start with −45 = −45; divide both sides by −9. −45 = −45 , −9 −9 or 5 = 5✓

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6–7 Discussion Groups, Sharing and Responding, Think-Pair-Share After students complete Item 6, they can discuss the process of solving this problem in Item 7. If there are students within a group having trouble getting from one step to the next in the solution, other students may provide an explanation. Additionally, these steps are samples, so if any students approach the problem with a different strategy (for example, they multiply through first by 100 to eliminate some of the decimals

and/or they multiply both sides of the equation by 4 in order to eliminate fractions), they could share this with their peers and demonstrate that there is more than one way to arrive at the correct solution.



Differentiating Instruction

4. Using the expression you wrote in Item 3, write an expression for the number of tubes the riders taking slide 2 will need, assuming each tube is full.


ELL

continued

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ACTIVITY 1 continued

8. Make sense of problems. Consider the meaning of the solution from Item 7. a. Explain why you should or should not round the value of r to the nearest whole number.

My Notes

Because the value of r represents a number of riders, and it doesn’t make sense to have a fraction of a rider, you should round r to the nearest whole number.

b. How many people out of the 100 riders will take slide 1?

Developing Math Language

46 riders

9. Use the expression you wrote in Item 2 to determine how long it will take the number of riders from Item 8b to go through slide 1. a. Evaluate the expression for the appropriate value of r.

( 2r )0.75 = ( 462 )0.75 = 17.25

MATH TIP When you evaluate an algebraic expression, you substitute values for the variables and then simplify the expression.

b. How many minutes will it take the riders to go through slide 1? Round to the nearest minute.



17 minutes

The rest of the 100 riders will go through slide 2 in about the same amount of time. So, your answer to Item 9b gives an estimate of the number of minutes it will take all 100 riders to go down the Penguin slides. 10. Recall that when a rider reaches the sign, there will be approximately 100 people waiting in front of him or her. What number should Jaabir write to complete the statement on the sign? From this point, your wait is approximately 17

8 Discussion Groups, Construct an Argument Have students work with a partner or in a small group to interpret the solution of r = 45.5. Have them explore whether this solution actually makes sense, given that r represents a number of riders (people). Is this a situation where rounding to the nearest whole number is necessary? If so, then why? Ask students to reflect on whether their results or solutions to a problem make sense.

minutes.


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This lesson contains the vocabulary term evaluate. When you evaluate an expression, you replace each variable in the expression by a given value and simplify the result. The term evaluate will surface again in later chapters, when students will be asked to evaluate the value of a function with a given input value. 9 Critique Reasoning, Discussion Groups, Construct an Argument It is important that students understand that the number of minutes written on the sign is an estimate, for several reasons. It is not certain that there will be exactly 100 people in front of the sign. There are different factors that could affect this number. For example, whether people stand close together or farther apart could impact the total number. Also, the size (age and weight) of the people would have an impact on the number of people in line. This problem depends upon approximations and estimations. Furthermore, the answer in Item 9, 17.25 minutes, does not divide evenly by the 1.25-minute timing cycle of slide 2 (neither does 17 minutes; the “common multiples” of 0.75 and 1.25 minutes closest to 17 are 15 and 18.75 minutes). 10–11 Discussion Groups, Construct an Argument, Debriefing Prior to Item 10, students read that it will take about the same amount of time for the remaining riders to go down slide 2. Since the result from Item 9b is 17 minutes, Jaabir should place 17 minutes on the board as the best approximation.


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ACTIVITY 1 continued My Notes

11. Describe how you could check that your answer to Item 10 is reasonable.

Sample answer: I could substitute the value of r into the equation and check that it makes the equation true. I could also evaluate the expression from Item 5 to check that I get the same time for slide 2 as for slide 1.

Check Your Understanding

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to interpreting the solution to an equation. When students answer Item 12, you may also want to have them explain how they arrived at their answer.

12. Suppose that Jaabir needs to place a second sign in the waiting line for the Penguin slides. When a rider reaches this sign, there will be approximately 250 people in front of him or her. What number should Jaabir write to complete the statement on this sign? Explain how you determined your answer.

Answers

13. Explain the relationships among the variable, expression, and equation.

ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning.

minutes.

LESSON 1-1 PRACTICE Use this information for Items 14–15. When full, one of the pools at Sapphire Island will hold 43,000 gallons of water. The pool currently holds 20,000 gallons of water and is being filled at a rate of 130 gallons per minute. 14. Write an equation that can be used to find h, the number of hours it will take to fill the pool from its current level. Explain the steps you used to write your equation.

ACADEMIC VOCABULARY When you interpret a solution, you state the meaning of the solution in the context of the problem or real-world situation.

15. Solve your equation from Item 14, and interpret the solution. Use this information for Items 16–18. Sapphire Island is open 7 days a week. The park has 8 ticket booths, and each booth has a ticket seller from 10:00 a.m. to 5 p.m. On average, ticket sellers work 30 hours per week. 16. Model with mathematics. Write an equation that can be used to find t, the minimum number of ticket sellers the park needs. Explain the steps you used to write your equation. 17. Solve your equation from Item 16, and interpret the solution. 18. The park plans to hire 20 percent more than the minimum number of ticket sellers needed in order to for sickness, vacation, and lunch breaks. How many ticket sellers should the park hire? Explain.

See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. LESSON 1-1 PRACTICE

14. 130(60h) + 20,000 = 43,000; Explanations will vary. Students may note that the number of gallons added to the pool plus the 20,000 gallons already in the pool must equal 43,000 gallons. They may also note that the minutes needed to fill the pool equals the number of hours h times 60.

ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving equations as well as interpreting solutions of equations. If students are having difficulty creating equations that model the situation, have them write out word equations.

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15. h ≈ 2.9; The solution shows that it will take about 3 more hours to fill the pool. 16. 30t = 49(8) or equivalent; Explanations may vary. Students may note that the minimum number of ticket sellers times 30 hours per week equals the total number of hours worked by the ticket sellers per week. Students may also note that the total number of hours worked by the ticket sellers each week is equal to the number of hours the park is open times the number of booths, or 49(8).

17. t ≈ 13.1; Some students may reason that 13.1 is sufficiently close to 13 that 13 ticket sellers will be enough for the park. Other students will note that the park technically needs more than 13 ticket sellers and will give an answer of 14. 18. 16 or 17, depending on the answer to Item 17



12. 43 min; Explanations may vary. Some students may write and solve an equation to find the answer. Other students may reason that the wait time for 250 riders will be 2.5 times the wait time for 100 riders. 13. Sample answer: A variable is a letter or symbol that represents an unknown value or values. An expression can include variables, along with numbers and operations. An equation is a statement that two expressions are equal.

From this point, your wait is approximately


10–11 (continued) Another way to check the answer would be to take 54 (which is the value of 100 − r, or the approximate number of slide 2 riders) and multiply it by 1.25; 54 × 1.25 = 16.875 ≈ 17 minutes. Alert students that even though each tube on slide 1 will hold 2 engers and each tube on slide 2 will hold 4 engers, the water park may not necessarily be filling every spot in each tube.




Lesson 1-2 Two-Variable Equations


Learning Targets:

equations in two variables to represent relationships between • Create quantities. • Graph two-variable equations.

PLAN

My Notes

Pacing: 1 class period Chunking the Lesson #1–2 #3–7 #8–13 #14 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Summarizing, Paraphrasing, Look for a Pattern, Think-Pair-Share, Create Representations, Interactive Word Wall, Identify a Subtask

At Sapphire Island, visitors can rent inner tubes to use in several of the park’s rides and pools. Maria works at the rental booth and is preparing materials so that visitors and employees will understand the pricing of the tubes. Renting a tube costs a flat fee of $5 plus an additional $2 per hour.

TEACH Bell-Ringer Activity Have students write each equation in the form Ax + By = C, where A, B, and C are integers and A is nonnegative.

As you work in groups on Items 1–7, review the above problem scenario carefully and explore together the information provided and how to use it to create potential solutions. Discuss your understanding of the problems and ask peers or your teacher to clarify any areas that are not clear. 1. Maria started making a table that relates the number of hours a tube is rented to the cost of renting the tube. Use the information above to help you complete the table.



Tube Rentals Hours Rented

Cost ($)

1

7

2

9

3

11

4

13

5

15

1. 4x − 7 = 2y [4x − 2y = 7] 2. 12 = y − 8x [8x − y = −12] 3. 5y = x − 3 [x − 5y = 3]

DISCUSSION GROUP TIP If you need help in describing your ideas during group discussions, make notes about what you want to say. Listen carefully to other group as they describe their ideas, and ask for clarification of meaning for any words routinely used by group .

2. Explain how a customer could use the pattern in the table to determine the cost of renting a tube for 6 hours. Sample answer: Add $2 to the cost of renting a tube for 5 hours: $15 + $2 = $17.

Next, Maria wants to write an equation in two variables, x and y, that employees can use to calculate the cost of renting a tube for any number of hours. 3. Reason abstractly. What does the independent variable x represent in this situation? Explain. The number of hours the tube is rented; The time in hours is the independent variable because this value determines the cost of renting the tube.

Lesson 1-2

MATH TIP Recall that in a relationship between two variables, the value of the independent variable determines the value of the dependent variable.


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1–2 Activating Prior Knowledge, Look for a Pattern, Role Play, Group Presentation Have students work in pairs or small groups to complete the second column of the table. Encourage them to find a pattern from one number to the next, going downward in both columns of the table. While they are working on this, circulate around the room and have students tell you the rental fee for some arbitrary number of hours that is not already in the table. After visiting various groups, have the students share their patterns and any other findings with the class as a whole. 3–7 Activating Prior Knowledge, Debriefing, Chunking the Activity, KWL Chart Discuss with students what it means to be independent vs. dependent in the real world. For example, students are dependents of their parents. Construct a KWL Chart by writing Know, Want to Know, and Learn as column headings in one row across the board. Beneath the Know column, write the Dependent variable and Independent variable, and ask students to define them in their own words, drawing from what they from their previous math courses.


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The cost in dollars of renting the tube; The cost is the dependent variable because this value depends on the number of hours the tube is rented.

5. Write an equation that models the situation. y = 5 + 2x

Technology Tip Students can use the table function on a graphing calculator to find how much to charge a customer, y, based upon the number of hours, x, as follows: Press y= to enter the equation 5 + 2x. Press 2nd WINDOW to access TBLSET. Set the Tblstart = 0, Tbl = 1, and Indpnt: to “Ask.” Then press 2nd GRAPH to access the table. At this screen, students can type in any number of hours at the X= prompt. The corresponding charge will appear next to it in the y1 column. This is a great way to check their work and practice using technology.

6. How can you tell whether the equation you wrote in Item 5 correctly models the situation?

Sample answer: Substitute values for x into the equation and check whether they give the correct values for y by comparing the results to the table on the previous page.

7. Construct viable arguments. Explain how an employee could use the equation to determine how much to charge a customer.

MATH TIP Before you can graph the equation, you need to determine the coordinates of several points that lie on its graph. One way to do this is by using pairs of corresponding values from the table on the previous page. You can also choose several values of x and substitute them into the equation to determine the corresponding values of y.

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Maria also thinks it would be useful to make a graph of the equation that relates the time in hours a tube is rented and the cost in dollars of renting a tube. 8. List five ordered pairs that lie on the graph of the relationship between x and y. Sample answer: (1, 7), (2, 9), (3, 11), (4, 13), (5, 15)




Sample answer: The employee should substitute the number of hours the customer rented the tube for x in the equation. Next, the employee should solve the equation for y. This value is the amount to charge the customer in dollars.

For additional technology resources, visit SpringBoard Digital.

8

4. What does the dependent variable y represent in this situation? Explain.


3–7 (continued) Beneath the Want to Know column, write how the dependent and independent variables are represented in this situation, as well as a linear equation that can be written to model the situation (refer to Bell-Ringer Activity if needed). Beneath the Learn column, write how the items in the first two columns can be tied together to answer Items 6 and 7.

003-016_SB_AG2_SE_U01_A01.indd Page 9 13/01/15 11:18 AM ehi-6





9. Use the grid below to complete parts a and b. a. Write an appropriate title for the graph based on the real-world situation. Also write appropriate titles for the x- and y-axes. b. Graph the ordered pairs you listed in Item 8. Then connect the points with a line or a smooth curve.

My Notes

Check students’ answers.

Tube Rentals

Cost ($)

y 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Universal Access

1

2 3 4 5 Number of Hours Rented

6

Yes, this is a linear equation. Sample explanation: The graph has a constant rate of change and therefore models a linear equation.


A common error students make is not connecting, or drawing a line through, the points. Emphasize that there are infinite solutions to equations in two variables, and that every point on the line is a solution.

x

10. Based on the graph, explain how you know whether the equation that models this situation is or is not a linear equation.


8–13 Predict and Confirm, Debriefing Have students predict what they think the graph will look like based upon the previous items leading up to this set of items. Remind students that a solution of an equation in two variables is an ordered pair of numbers, which can be plotted on a coordinate plane. The easiest way to get these ordered pairs is from the table created in Item 1. The axes should be labeled according to what the x- and y-values represent— hours and cost, respectively. When graphed, the ordered pairs should form a straight line. In this particular case, the graph is in the first quadrant because you cannot rent for a negative number of hours. Item 12 refers to the y-intercept; explain why there is no x-intercept (at least for this situation).

MATH TIP

11. Reason quantitatively. Explain why the graph is only the first quadrant.

Recall that a linear equation is an equation whose graph is a line. A linear equation can be written in standard form Ax + By = C, where A, B, and C are integers and A is nonnegative.

12. What is the y-intercept of the graph? Describe what the y-intercept represents in this situation.

MATH TIP

The variable x represents time and the variable y represents cost. It would not make sense for either of these variables to be negative.

5; the flat fee in dollars for renting a tube

13. What is the slope of the graph? Describe what the slope represents in this situation. 2; the hourly cost in dollars of renting a tube

The y-intercept of a graph is the y-coordinate of a point where the graph intersects the y-axis. The slope of a line is the ratio of the change in y to the change in x between any two points.


9


9






DISCUSSION GROUP TIP Share your description with your group and list any details you may not have considered before. If you do not know the exact words to describe your ideas, use synonyms or request assistance from group to help you convey your ideas. Use nonverbal cues such as raising your hand to ask for clarification of others’ ideas.

14. Work with your group. Describe a plausible scenario related to the water park that could be modeled by this equation: y = 40x − 8. In your description, be sure to use appropriate vocabulary, both real-world and mathematical. Refer to the Word Wall and any notes you may have made to help you choose words for your description. Sample answer: The water park mails out coupons for $8 off the total cost of a ticket purchase. Tickets to the park normally cost $40 each. The equation models the cost y in dollars of purchasing x tickets to the water park with a coupon.

Check Your Understanding 15. Explain why the slope of the line you graphed in Item 9 is positive. 16. Explain how you would graph the equation from Item 14. What quantity and units would be represented on each axis?

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to two-variable linear equations.

17. Is the equation y = −2x + x2 a linear equation? Explain how you know.

Answers

LESSON 1-2 PRACTICE

15. The cost of renting a tube increases as the number of hours it is rented increases. So, y increases as x increases, which indicates a positive slope. 16. Sample answer: Substitute several values of x into the equation to find the corresponding values of y. Then use the values of x and y to write ordered pairs. Finally, graph the ordered pairs and draw a line through them. Quantities and units will vary depending on the scenario the student wrote for Item 14. 17. No. Sample explanation: The equation cannot be written in the form Ax + By = C. The graph of the equation is a curve, not a line.

Use this information for Items 18–22. Some of the water features at Sapphire Island are periodically treated with a chemical that prevents algae growth. The directions for the chemical say to add 16 fluid ounces per 10,000 gallons of water.

19. Write a linear equation in two variables that models the situation. Tell what each variable in the equation represents. 20. Graph the equation. Be sure to include titles and use an appropriate scale on each axis. 21. What are the slope and y-intercept of the graph? What do they represent in the situation? 22. Construct viable arguments. An employee adds 160 fluid ounces of the chemical to a feature that holds 120,000 gallons of water. Did the employee add the correct amount? Explain.

ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity.

ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and graphing equations in two variables. If students are having difficulty graphing the equations, review the process of creating a table of values and identifying slope and intercepts.

10

10


LESSON 1-2 PRACTICE

18.

Amount of Chemical to Add Gallons of Water

Fluid Ounces of Chemical

10,000

16

20,000

32

30,000

48

40,000

64

50,000

80

19. Sample answer: y = 1.6x; y represents the number of fluid ounces of the chemical to add to a water feature, and x represents the amount of water the feature holds, in thousands of gallons.



18. Make a table that shows how much of the chemical to add for water features that hold 10,000; 20,000; 30,000; 40,000; and 50,000 gallons of water.


14 Marking the Text, Debriefing, Discussion Groups, Group Presentation Write the slope-intercept form of a line (y = mx + b) on the board. Write the equation y = 40x − 8 right below. Highlight that the slope of the equation, m, is 40. Highlight that the y-intercept, b, is −8. Be sure to stress this is a negative value this time. Have students spend a few minutes collaborating in small groups to create a plausible scenario, related to the water park, for which this equation could be modeled. Ask students to share their ideas and have a classroom discussion as to why their scenarios are or are not plausible.




Lesson 1-3 Absolute Value Equations and Inequalities


PLAN

My Notes

Learning Targets:

• Write, solve, and graph absolute value equations. • Solve and graph absolute value inequalities.

Pacing: 1 class period Chunking the Lesson

SUGGESTED LEARNING STRATEGIES: Marking the Text, Interactive Word Wall, Close Reading, Create Representations, Think-Pair-Share, Identify a Subtask, Quickwrite, Self Revision/Peer Revision

You can use the definition of absolute value to solve absolute value equations algebraically. Since

−(ax + b) if ax + b < 0 ax + b =  ,  ax + b if ax + b ≥ 0 then the equation |ax + b| = c is equivalent to −(ax + b) = c or (ax + b) = c. Since −(ax + b) = c is equivalent to ax + b = −c, the absolute value equation |ax + b| = c is equivalent to ax + b = −c or ax + b = c.

Example A

2|4 − 1| − 5 = 1 2|3| − 5 = 1 2(3) − 5 = 1 6−5=1

An absolute value equation is an equation involving the absolute value of a variable expression.

MATH TIP Recall that the geometric interpretation of |x| is the distance from the number x to 0 on a number line. If |x| = 5, then x = −5 or x = 5 because those two values are both 5 units away from 0 on a number line.

1

2

3

4

5

d. |x + 2| + 3 = 1 no solution

LESSON 1-3 PRACTICE (continued)

20.

Amount of Chemical to Add y 96 80 64 48 32 16 10

20

30

40

50

Amount of Water (1000 gal)

x

1. |6| [6] 2. |−6| [6] Then have students solve the following equation. [x = 6 or x = −6]

An absolute value equation is an equation involving an absolute value of an expression containing a variable. Just like when solving algebraic equations without absolute value bars, the goal is to isolate the variable. In this case, isolate the absolute value bars because they contain the variable. It should be emphasized that when solving absolute value equations, students must think of two cases, as there are two numbers that have a specific distance from zero on a number line.

x = 1, x = −3

x=3

Bell-Ringer Activity Students should recall that an absolute value of a number is its distance from zero on a number line.


Solve each absolute value equation. Graph the solutions on a number line. a. |x − 2| = 3 b. |x + 1| − 4 = −2 c. |x − 3| + 4 = 4

TEACH

Example A Marking the Text, Interactive Word Wall Point out the Math Tip to reinforce why two solutions exist. Work through the solutions to the equation algebraically. Remind students that solutions to an equation make the equation a true statement. This mathematical understanding is necessary for students to be able to check their results.

Try These A x = 5, x = −1

#2

3. |x|= 6

4 0

Example C

Example B

Have students evaluate the following:

To graph the solutions, plot points at 4 and −2 on a number line. –5 –4 –3 –2 –1

#1

Lesson Practice

MATH

2|−2 − 1| − 5 = 1 2|−3| − 5 = 1 2(3) − 5 = 1 6−5=1

–2

Example A


Check to see if both solutions satisfy the original equation. Substitute 4 and −2 for x in the original equation.

Amount of Chemical (fl oz)



Solve 2|x − 1| − 5 = 1. Graph the solutions on a number line. Step 1: Isolate the absolute value 2|x − 1| − 5 = 1 expression. Add 5 to both 2|x − 1| = 6 sides and then divide by 2. |x − 1| = 3 Step 2: Write and solve two equations x − 1 = 3 or x − 1 = −3 x = 4 or x = −2 using the definition of absolute value. x = 4 and x = −2 Solution: There are two solutions:

Lesson 1-3


21. The slope is 1.6; the slope represents the number of fluid ounces of the chemical to add to 1000 gallons of water. The y-intercept is 0; the y-intercept shows that no chemical should be added when a feature contains no water. 22. No. Sample explanation: Substituting 120 for x in the equation and solving for y shows that the employee should have added 192 fluid ounces of the chemical.

11


11






There are possibly two, one, or zero solutions to an absolute value equation having this form.

Have students look back at Try These A, parts c and d. Have volunteers construct a graph of the two piecewise-defined functions used to write each equation and then discuss how the solution set is represented by the graph.

Example B The temperature of the wave pool at Sapphire Island can vary up to 4.5°F from the target temperature of 82°F. Write and solve an absolute value equation to find the temperature extremes of the wave pool. (The temperature extremes are the least and greatest possible temperatures.) Step 1: Write an absolute value equation to represent the situation.

You know that the distance from t to 82°F on a thermometer is 4.5°F. This distance can be modeled with the absolute value expression |t − 82|.

Use the definition of absolute value to solve for t. |t − 82| = 4.5 t − 82 = 4.5 or t − 82 = −4.5 t = 86.5 or t = 77.5 Solution: The greatest possible temperature of the wave pool is 86.5°F, and the least possible temperature is 77.5°F. Both of these temperatures are 4.5°F from the target temperature of 82°F.

The pH of water is a measure of its acidity. The pH of the water on the Seal Slide can vary up to 0.3 from the target pH of 7.5. Use this information for parts a–c. a. Write an absolute value equation that can be used to find the extreme pH values of the water on the Seal Slide. Be sure to explain what the variable represents. |p − 7.5| = 0.3; p represents the extreme pH values of the water.

b. Solve your equation, and interpret the solutions.

For those students for whom English is a second language, explain that the word varies in mathematics means changes. There are different ways of thinking about how values can vary. Values can vary upward or downward, less than or greater than, in a positive direction or a negative direction, and so on. However, the importance comes in realizing that there are two different directions, regardless of how you think of it.

12

Step 2:

Try These B

ELL

Also address the word extremes as it pertains to mathematics. An extreme value is a maximum value if it is the largest possible amount (greatest value), and an extreme value is a minimum value if it is the smallest possible amount (least value).

Let t represent the temperature extremes of the wave pool in degrees Fahrenheit. |t − 82| = 4.5

MATH TIP

p = 7.8, p = 7.2; The greatest allowed pH is 7.8, and the least allowed pH is 7.2.

c. Reason quantitatively. Justify the reasonableness of your answer to part b. Sample answer: 7.8 is 0.3 more than 7.5; 7.2 is 0.3 less than 7.5.

12




Example B Marking the Text, Simplify the Problem, Critique Reasoning, Group Presentation Start with emphasizing the word vary in the Example, discussing what it means when something varies. You may wish to present a simpler example such as: The average cost of a pound of coffee is $8. However, the cost sometimes varies by $1. This means that the coffee could cost as little as $7 per pound or as much as $9 per pound. Now have students work in small groups to examine and solve Example B by implementing an absolute value equation. Additionally, ask them to take the problem a step further and graph its solution on a number line. Have groups present their findings to the class.

1. Reason abstractly. How many solutions are possible for an absolute value equation having the form |ax + b| = c, where a, b, and c are real numbers?


1 Identify a Subtask, Quickwrite When solving absolute value equations, students may not see the purpose in creating two equations. Reviewing the definition of the absolute value function as a piecewise-defined function with two rules may enable students to see the reason why two equations are necessary.






Solving absolute value inequalities algebraically is similar to solving absolute value equations. By the definition of absolute value, |ax + b| > c, where c > 0, is equivalent to −(ax + b) > c or ax + b > c. Multiplying the first inequality by −1, and then using a similar method for |ax + b| < c, gives these statements:

MATH An absolute value inequality is an inequality involving the absolute value of a variable expression.

• |ax + b| > c, c > 0, is equivalent to ax + b < −c or ax + b > c. • |ax + b| < c, c > 0, is equivalent to ax + b < c or ax + b > −c, which can also be written as −c < ax + b < c.

Example C Solve each inequality. Graph the solutions on a number line. a. |2x + 3| + 1 > 6 Step 1: Isolate the absolute |2x + 3| + 1 > 6 value expression. |2x + 3| > 5 Step 2: Write two inequalities. 2x + 3 > 5 or 2x + 3 < −5 Step 3: Solve each inequality. x > 1 or x < −4 Solution: 0

–4

b. |3x − 1| + 5 < 7 Step 1: Isolate the absolute value expression. Step 2:

Write the compound inequality.

Step 3:

Solve the inequality.


These properties of real numbers can help you solve inequalities. The properties also apply to inequalities that include <, ≥, or ≤.

1

Addition Property of Inequality If a > b, then a + c > b + c. Subtraction Property of Inequality If a > b, then a − c > b − c.

|3x − 1| + 5 < 7 |3x − 1| < 2 −2 < 3x − 1 < 2

Try These C Solve and graph each absolute value inequality. b. x + 2 − 3 ≤ −1 −4 ≤ x ≤ 0 a. x − 2 > 3 x > 5 or x < −1 5

–1

c. 5x − 2 + 1 ≥ 4 x ≥ 1 or x ≤ − 1 5

–1 5

–6

See graph A.

For example: |x| < 5; this means the value of the variable x is less than 5 units away from the origin (zero) on a number line. The solution is −5 < x < 5. See graph B. Students can apply these generalizations to Example C. Point out that they should proceed to solve these just as they would an algebraic equation, except in two parts, as shown above. After they have some time to work through parts a and b, discuss the solutions with the whole class.

d. 2 x + 7 − 4 < 1 −6 < x < −1

1

For example, |x| > 5 means the value of the variable x is more than 5 units away from the origin (zero) on a number line. The solution is x < − 5 or x > 5.

This also holds true for |A| ≤ b.

0

–4

Example C Simplify the Problem, Debriefing Before addressing Example C, discuss the following: Inequalities with |A| > b, where b is a positive number, are known as disjunctions and are written as A < −b or A > b.

Inequalities with |A| < b, where b is a positive number, are known as conjunctions and are written as −b < A < b, or as −b < A and A < b.

Division Property of Inequality If a > b and c > 0, then a > b . c c If a > b and c < 0, then a < b . c c

1

–1 3

An absolute value inequality is basically the same as an absolute value equation, except that the equal sign is now an inequality symbol: <, >, ≤, ≥, or ≠. It still involves an absolute value expression that contains a variable, just like before. Use graphs on a number line of the solutions of simple equations and inequalities and absolute value equations and inequalities to show how these are all related.

This also holds true for |A| ≥ b.

Multiplication Property of Inequality If a > b and c > 0, then ca > cb. If a > b and c < 0, then ca < cb.

−1 < x <1 3

Solution:


MATH TIP


–1

TEACHER to TEACHER


A

–7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

–7 –6 –5 –4 –3 –2 –1

0

1

2

3

4

5

6

7

B

13

Another method for solving inequalities relies on the geometric definition of absolute value |x − a| as the distance from x to a. Here’s how you can solve the inequality in the example: |2x + 3| + 1 > 6 |2x + 3| > 5 2 x − −3 > 5 2 x − −3 > 5 2 2 Thus, the solution set is all values of x whose distance from − 3 is greater 2 than 5 . The solution can be 2 represented on a number line and written as x < −4 or x > 1.


13






2. Make sense of problems. Why is the condition c > 0 necessary for |ax + b| < c to have a solution? If c = 0 or if c < 0 (c is negative), the inequality would be impossible or trivial because absolute value is the distance from zero on a number line.

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to absolute value equations. Have groups of students present their solutions to Item 4. 3. Sample answer: A linear equation of the form ax + b = c has 1 solution, but an absolute value equation of the form |ax + b| = c may have 0, 1, or 2 solutions depending on the value of c. 4. a. She did not isolate the absolute value expression first. b. Sample answer: She could have substituted the values for x into the original equation to check whether they satisfy the equation. c. No solution; Sample explanation: First, isolate the absolute value expression: |x + 5| = −4. The equation |x + 5| = −4 has no solutions because the absolute value of an expression cannot be negative. 5. Sample answer: The inequality has the form |ax − b| ≥ c, where c ≥ 0, so it can be written as ax − b ≥ c or ax − b ≤ −c. Thus, |5x − 6| ≥ 9 is equivalent to 5x − 6 ≥ 9 or 5x − 6 ≤ −9.

ACADEMIC VOCABULARY When you compare and contrast two topics, you describe ways in which they are alike and ways in which they are different.

3. Compare and contrast a linear equation having the form ax + b = c with an absolute value equation having the form |ax + b| = c. 4. Critique the reasoning of others. Paige incorrectly solved an absolute value equation as shown below. −2 |x + 5| = 8 −2(x + 5) = 8 or −2(x + 5) = −8 −2x − 10 = 8 or −2x − 10 = −8 x = −9 or x = −1 a. What mistake did Paige make? b. How could Paige have determined that her solutions are incorrect? c. Solve the equation correctly. Explain your steps. 5. Explain how to write the inequality |5x − 6| ≥ 9 without using an absolute value expression.

LESSON 1-3 PRACTICE Solve each absolute value equation. 6. |x − 6| = 5

7. |3x − 7| = 12

8. |2x + 9| − 10 = 5

9. |5x − 3| + 12 = 4

10. Model with mathematics. The flow rate on the Otter River Run can vary up to 90 gallons per minute from the target flow rate of 640 gallons per minute. Write and solve an absolute value equation to find the extreme values of the flow rate on the Otter River Run. Solve each absolute value inequality. Graph the solutions on a number line.

ASSESS

11. |x − 7| > 1

12. |2x − 5| ≤ 9

13. |3x − 10| − 5 ≥ −1

14. |4x + 3| − 9 < 5

Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity.

ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving absolute value equations and inequalities and graphing the solutions of absolute value equations and inequalities. If students are still having difficulty, review the process of rewriting an absolute value equation or inequality as two equations or inequalities.

14

x = 1, x = 11 x = 19 , x = − 5 3 3 x = 3, x = −12 no solution | f − 640| = 90, where f represents the extreme flow rates in gallons per minute; f = 730, f = 550; The greatest flow rate is 730 gallons per minute and the least flow rate is 550 gallons per minute. 11. x < 6 or x > 8 6. 7. 8. 9. 10.

0

14


LESSON 1-3 PRACTICE

2

4

6

12. −2 ≤ x ≤ 7

–4 –2

0

2

4

13. x ≤ 2 or x ≥ 14 3

0

2

4

6

–4 –2

0

2

4

–2

14. − 17 < x < 11 4 4

8


6

8


Answers



2 Quickwrite, Self Revision/Peer Revision, Debriefing Use the investigation regarding the restriction c > 0 as an opportunity to discuss the need to identify impossible situations involving inequalities.




Creating Equations One to Two


Jerome bought a sweater that was on sale for 20 percent off. Jerome paid $25.10 for the sweater, including sales tax of 8.25 percent. Use this information for Items 7–9.

ACTIVITY 1 PRACTICE


Lesson 1-1

7. Write an equation that can be used to find the original price of the sweater.

Susan makes and sells purses. The purses cost her $12 each to make, and she sells them for $25. This Saturday, she is renting a booth at a craft fair for $60. Use this information for Items 1–3.

8. Solve the equation, and interpret the solution. 9. How much money did Jerome save by buying the sweater on sale? Explain how you determined your answer.

1. Write an equation that can be used to find the number of purses Susan must sell to make a profit of $250 at the fair.

Lesson 1-2

2. Solve the equation, and interpret the solution.

A taxi company charges an initial fee of $3.50 plus $2.00 per mile. Use this information for Items 10–16.

3. If Susan sells 20 purses at the fair, will she meet her profit goal? Explain why or why not.

10. Make a table that shows what it would cost to take a taxi for trips of 1, 2, 3, 4, and 5 miles.

A medical rescue helicopter is flying at an average speed of 172 miles per hour toward its base hospital. At 2:42 p.m., the helicopter is 80 miles from the hospital. Use this information for Items 4–6.

11. Write an equation in two variables that models this situation. Explain what the independent variable and the dependent variable represent.

4. Which equation can be used to determine m, the number of minutes it will take the helicopter to reach the hospital? A. 172(60m) = 80 B. 172 m = 80 60

12. Graph the equation. Be sure to include a title for the graph and for each axis. 13. Describe one advantage of the graph compared to the equation.

( )

14. Is the equation that models this situation a linear equation? Explain why or why not.

( )

15. What are the slope and y-intercept of the graph? What do they represent in the situation?

6. An emergency team needs to be on the roof of the hospital 3 minutes before the helicopter arrives. It takes the team 4 minutes to reach the roof. At what time should the team start moving to the roof to meet the helicopter? Explain your reasoning.

16. Shelley uses her phone to determine that the distance from her apartment to Blue Café is 3.7 miles. How much it will cost Shelley to take a taxi to the café?

10.

Taxi Fare Trip Length (mi)

Cost ($)

1

5.50

2

7.50

3

9.50

4

11.50

5

13.50

11. y = 3.5 + 2x; The independent variable x represents the length of the trip in miles. The dependent variable y represents the cost of the trip in dollars. 12. Taxi Fare y

14

19. y

Yearly Expenses of Activity 1 • Creating Equations Large-Cat Exhibit

12

15

10

Cost ($)

14. Yes. Sample explanation: The graph of the equation is a line. 15. The slope is 2. It represents the $2.00 charge per mile for taking a taxi. The y-intercept is 3.5. It represents the initial fee of $3.50 for taking a taxi. 16. $10.90 (not including tip) 17. B 18. y = 8000x + 6000(8 − x) or equivalent

64

Yearly Expenses ($1000)



C. 172 60 = 80 D. 172 = 80 m 60m 5. Solve the equation, and interpret the solution.

ACTIVITY PRACTICE 1. 25p − 12p − 60 = 250 or equivalent, where p is the number of purses sold 2. p ≈ 23.8; Susan must sell 24 purses to make a profit of $250. 3. No. Sample explanation: Susan must sell at least 24 purses to meet her profit goal, and 20 is less than 24. 4. B 5. m ≈ 27.9; It will take the helicopter about 28 minutes to reach the hospital. 6. 3:03 p.m.; Sample explanation: The helicopter will reach the hospital 28 minutes after 2:42 p.m., or at 3:10 p.m. The team needs to start moving 7 minutes before the helicopter arrives, or at 3:03 p.m. 7. 0.80p + 0.0825(0.80p) = 25.10 or equivalent, where p is the original price in dollars 8. p ≈ 28.98; The original cost of the sweater is $28.98. 9. $6.27; Sample explanation: The cost of the original sweater with tax is $28.98 + 0.0825($28.98) = $31.37. The difference between the original cost of the sweater with tax and the sale price of the sweater with tax is $31.37 − $25.10 = $6.27.

60

8 6 4

56

2

52 1

48

2

3

4

x

5

Trip Length (mi)

44 1 2 3 4 5 6 7 8

Number of Lions

x

13. Sample answer: The graph makes it easy to see that as the trip length increases, the cost also increases.


15



2

–6

–7 –6 –5 –4 –3 –2 –1 0 1 2 3

25. |t − 101.7| = 0.6 or equivalent, where t represents the extreme possible values of Zachary’s actual temperature in °F; t = 102.3, t = 101.1; Zachary’s actual temperature is between 101.1°F and 102.3°F. 26. Check students’ graphs. a. −17 < x < 7 b. x ≥ 11 or x ≤ −3 5 c. 2 ≤ x ≤ 2 5 d. x > 12 or x < 2 e. no solutions 27. C 28. a. Lesley’s Cell Phone y

Creating Equations One to Two


17. Choose the equation that is not linear. A. y = 2 x + 6 B. y = 4 − 1 3 x C. 3x + 2y = 8 D. x = −4 A zoo is building a new large-cat exhibit. Part of the space will be used for lions and part for leopards. The exhibit will house eight large cats in all. Expenses for a lion will be about $8000 per year, and expenses for a leopard will be about $6000 per year. Use this information for Items 18–21. 18. Write an equation that can be used to find y, the yearly expenses for the eight cats in the exhibit when x of the cats are lions. 19. Graph the equation. Be sure to include a title for the graph and for each axis.

25. A thermometer is accurate to within 0.6°F. The thermometer indicates that Zachary’s temperature is 101.7°F. Write and solve an absolute value equation to find the extreme possible values of Zachary’s actual temperature. 26. Solve each absolute value inequality. Graph the solutions on a number line. a. |x + 5| < 12 b. |5x + 2| ≥ 13 c. |10x − 12| − 9 ≤ −1 d. |x − 7| + 3 > 8 e. |−2x + 5| + 6 ≥ 4 27. Which number line shows the solutions of the inequality 2|x − 1| ≥ 4?

20. Are all points on the line you graphed solutions in this situation? Explain. 21. What would the yearly expenses be if five of the cats in the exhibit are lions and the rest are leopards? Explain how you found your answer.

Lesson 1-3 22. Solve each absolute value equation. a. |2x − 3| = 7 b. |2x + 5| = 23 c. |x − 10| − 11 = 12 − 23 d. |7x + 1| − 7 = 3 e. |2x| − 3 = −5 23. If the center thickness of a lens varies more than 0.150 millimeter from the target thickness of 5.000 millimeters, the lens cannot be used. Write and solve an absolute value equation to find the extreme acceptable values for the center thickness of the lens. 24. Solve the equation |2x + 4| − 1 = 7. Then graph the solutions on a number line.

A. –2 –1

0

1

2

3

4

B. –2 –1

0

1

2

3

4

C. –2 –1

0

1

2

3

4

D. –2 –1

0

1

2

3

4

MATHEMATICAL PRACTICES Attend to Precision

28. The equation y = 0.5x + 40 represents the monthly cost y in dollars of Lesley’s cell phone, where x is the number of talk minutes over 750 that Lesley uses. a. Graph the equation. b. How did you determine the range of values to show on each axis of your graph? c. What are the units on each axis of your graph? d. What are the units of the slope of the linear equation? Explain. e. Write a different plausible scenario—not related to cell phone costs—that could be modeled using the equation y = 0.5x + 40. Be sure to use appropriate vocabulary, both real-world and mathematical.

Monthly Cost ($)

46 44

16

42 40 38 2

4

6

8

10

x

Talk Minutes over 750

ADDITIONAL PRACTICE If students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital for additional practice problems.

16


28. b. Sample answer: x represents a number of talk minutes, which means that x cannot be negative. y represents the monthly cost of the cell phone, which means that y cannot be negative, either. The least possible monthly cost of the cell phone is $40, so I put a break in the y-axis between 0 and 38 so that the graph would not need to be so tall. When x = 10, y = 45, so I let the y-axis go up to 48 to include all of the y-values as x increases from 0 to 10.

c. x-axis: minutes over 750; y-axis: dollars d. Dollars per minute over 750; Sample explanation: The slope is the ratio of the change in y to the change in x between any two points on the line, so the units of the slope are units of y per units of x. e. Answers may vary. Sample answer: The equation represents the amount of money Gary saves if he started with $40 and then saved $0.50 per day.



20. No; x represents a number of lions, so x must be a whole number. Also, no more than 8 lions can be in the exhibit, so the greatest value of x is 8. Thus, the only points on the line that are solutions are those with an x-coordinate that is a whole number no greater than 8. 21. $58,000; Sample explanation: The graph includes the point (5, 58). This point represents a yearly cost of $58,000 when 5 of the large cats in the exhibit are lions. 22. a. x = 5, x = −2 b. x = 9, x = −14 c. x = 10 d. x = 9 , x = − 11 7 7 e. no solutions 23. |t − 0.150| = 5.000 or equivalent, where t represents the extreme acceptable values for the center thickness of the lens; t = 5.150, t = 4.850; The greatest acceptable center thickness is 5.150 mm, and the least acceptable center thickness is 4.850 mm. 24. x = 2, x = −6



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ACTIVITY

Graphing to Find Solutions

ACTIVITY 2

Choices Lesson 2-1 Graphing Two-Variable Equations

In Activity 2, students represent constraints using equations and/or inequalities. They graph these constraints on a coordinate plane. Then they use their graphs to determine solutions to a system of equations or system of inequalities. Throughout this activity, emphasize the process of writing equations and inequalities from verbal descriptions and generating solutions once the constraints are graphed on the coordinate plane.

My Notes

equations in two variables to represent relationships between • Write quantities. • Graph equations on coordinate axes with labels and scales.

SUGGESTED LEARNING STRATEGIES: Shared Reading, Marking the Text, Graphic Organizer, Create Representations, Look for a Pattern, Group Presentation, Activating Prior Knowledge

Roy recently won a trivia contest. The prize was a five-day trip to New York City, including a round-trip airplane ticket and $3000 in cash. The money will pay the cost of a hotel room, meals, entertainment, and incidentals. To prepare for his trip, Roy gathered this information.

Lesson 2-1

• A hotel room in New York City costs $310 per night, and the trip includes staying five nights. • A taxi between New York City and LaGuardia Airport will cost $45 each way.

PLAN Pacing: 1 class period Chunking the Lesson

Roy must set aside the cash required to pay for his hotel room and for taxi service to and from the airport. Once he has done this, Roy can begin to make plans to enjoy the city with his remaining prize funds.

#1 #2–4 #5–6 #7 Check Your Understanding Lesson Practice

1. Reason quantitatively. How much money will Roy have available to spend on performances, meals, and any other expenses that might arise after paying for his hotel and taxis? Show your work.

#8–10

TEACH

$1360 Sample answer: 3000 − 5(310) − 2(45) = 1360


Investigative Activity Standards Focus

Learning Targets:


2

Bell-Ringer Activity Have students write a function for each situation. 1. cost of plumbing repairs: $35/hr initial fee for repair: $50 [C(h) = 50 + 35h] 2. descent of hot air balloon: 5 ft/min initial height of balloon: 250 ft [H(m) = 250 − 5m] 3. number of students: 18 per bus other students: 135 [S(b) = 135 + 18b] Discuss with students the method they used to write the functions and the definitions of the variables they chose.

During his trip to New York City, Roy wants to spend only his winnings from the contest. He wants to focus on two of his favorite pastimes: attending theater or musical performances and dining in restaurants. After surfing the web, Roy determines the following facts: • On average, a ticket for a performance in New York City costs $100. • He will spend on average $40 per meal.

1 Debriefing This item is designed as an entry-level question. It will be used throughout the activity, so a debriefing is important to make sure that all students have the correct answer as they progress through the activity.

Activity 2 • Graphing to Find Solutions

17

Common Core State Standards for Activity 2 HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. HSA-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.


17




Lesson 2-1 Graphing Two-Variable Equations

ACTIVITY 2 Items 2–10 are designed to review how to work with linear equations and activate students’ prior knowledge. It may be possible to work through these items quickly without using the mini-lessons on this page and the next. If students need more , take the time to have discussions about rate, y-intercept, slope, slope-intercept form, and point-slope form as needed.

My Notes

2–4 Create Representations, Look for a Pattern, Quickwrite Ensure that students know the difference between discrete and continuous data. In this case, only a whole number of tickets can be purchased, so the graph is individual points rather than a line. Watch for students that have 760, 660, and 560 as their last three entries in the table. These students may not realize that the numbers of tickets in the last three rows of the table are no longer increasing by 1 each time.

Tickets (t)

Money Available (M)

0

1360

1

1260

2

1160

3

1060

4

960

5

860

8

560

10

360

13

60

M 1400 1200 1000 800 600 400 200 5

10

t

Sample explanation: The rate of change is −100, so for each additional ticket purchased, subtract $100 from the amount available.

MATH TIP

18

5. Write a function M(t) that represents the amount of money that Roy has left after purchasing t tickets. M(t) = 1360 − 100t


MINI-LESSON: Slope-Intercept Form of the Equation of a Line If students need additional help with finding slope or finding the equation of a line in slope-intercept form, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.



4. Explain how you determined the values for 8, 10, and 13 tickets.


Sample answer: There is a linear pattern in both the graph and the table such that the amount of money available decreases by $100 for each ticket purchased.

A function is a relationship between two quantities in which each input has exactly one output.

18

15

Tickets

3. What patterns do you notice?

When students look for the pattern in Item 3, make sure they relate the fact that the amount of money decreasing represents a negative rate of change. 5 Create Representations, Group Presentation, Debriefing Ask students to share their answers. Review slope as a constant rate of change for linear functions. Have students look at successive differences in the table, and use the graph to review how to find the slope of a linear function.

2. Model with mathematics. Roy wants to know how the purchase of each ticket affects his available money. Fill in the table below. Plot the points on the grid.

Money Available ($)


continued






6. Use mathematical terminology to explain what −100 and 1360 each represent in your function in Item 5.

7 Create Representations Students will replicate what they did in Items 2–6 without the scaffolding. Use this time to help individual students who need more in the review process.

Sample explanation: The coefficient of t, −100, is the slope, and the constant 1360 is the y-intercept of the line.

7. Roy wonders how his meal costs will affect his spending money. a. Write a function D(m) that represents the amount of money Roy has left after purchasing m number of meals. D(m) = 1360 − 40m

b. Graph your function on the grid.

D(m)

CONNECT TO TECHNOLOGY 1400

You can also graph the function by using a graphing calculator. When entering the equation, use x for the independent variable and y for the dependent variable.

Money Left ($)

1200 1000 800

6 Think-Pair-Share, Quickwrite Ensure students make the connection between the constant term and the y-intercept of a line in slope-intercept form.

600

8–10 Look for a Pattern, Activating Prior Knowledge, Interactive Word Wall, Quickwrite, Debriefing Review the concept of domain and clarify the idea of the contextual domain. Discuss why negative values have no meaning in the situation.

CONNECT TO TECHNOLOGY For additional technology resources, visit SpringBoard Digital.

400 200 20

40

60

m

Meals

8. What kind of function is D(m)?



It is linear.

9. What is the rate of change for D(m), including units? −40

MATH TIP

$ , or −$40 per meal meal

10. Make sense of problems. Are all the values for m on your graph valid in this situation, given that m represents the number of meals that Roy can buy? Explain.

The rate of change of a function is the ratio of the amount of change in the dependent variable to the amount of change in the independent variable.

No. Sample explanation: The contextual domain is restricted to positive integers less than 35 because the number of meals will be a positive number and the amount of money left cannot be negative.


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MINI-LESSON: Point-Slope Form of the Equation of a Line If students need additional help with writing an equation using point-slope form, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.


19




ACTIVITY 2

Debrief students’ answers to these items to ensure that they understand concepts related to interpreting the rate of change and intercepts of an equation.

continued My Notes


Answers 11. The y-intercept of 1360 represents the amount of money in dollars Roy will have if he does not buy any meals. The x-intercept of 34 represents the greatest number of meals Roy can buy before running out of money. 12. Write the ratio of the change in the dependent variable to the change in the independent variable. 13. The rate of change of a linear function is the same as the slope. 14. Sample answer: Use the coordinates of the two points to find the slope of the line. Then graph the two points and draw a line through them to determine the y-intercept of the line. Finally, write the equation in slope-intercept form.

11. What do the x- and y-intercepts of your graph in Item 7 represent? 12. If you know the coordinates of two points on the graph of a linear function, how can you determine the function’s rate of change? 13. What is the relationship between the rate of change of a linear function and the slope of its graph?

MATH TIP The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.

14. Using your answers to Items 12 and 13, explain how to write the equation of a line when you are given the coordinates of two points on the line.

LESSON 2-1 PRACTICE

15. Write the equation of the line with y-intercept −4 and a slope of 3 . 2 Graph the equation.

16. Write the equation of the line that es through the point (−2, −3) and has a slope of 5. Graph the equation. 17. Model with mathematics. Graph the function f (x ) = 3 − 1 (x − 2) . 2 Use the following information for Items 18–20. Roy already has 10,368 frequent flyer miles, and he will earn 2832 more miles from his round-trip flight to New York City. In addition, he earns 2 frequent flyer miles for each dollar he charges on his credit card.


18. Write the equation of a function f(d) that represents the total number of frequent flyer miles Roy will have after his trip if he charges d dollars on his credit card. 19. Graph the function, using appropriate scales on the axes. 20. Reason quantitatively. How many dollars will Roy need to charge on his credit card to have a total of 15,000 frequent flyer miles? Explain how you determined your answer.


10

y


15. y = 3 x − 4 2

8 6 4 2 –10 –8 –6 –4 –2 –2

2

4

6

8

10

x

–4 –6

ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and graphing linear equations, as well as how to interpret rate of change and intercepts of equations. If students are continuing to having difficulty writing equations to model a given situation, have them practice writing word expressions and translating the word expressions into algebraic expressions.

20

20


16. y = 5x + 7 10

17.

y

10

6

6

4

4

2

2 2

4

6

8

10

x

–10 –8 –6 –4 –2 –2

4

6

–4

–4

–6

–6

–8

–8

–10

–10

2

18. f(d) = 13,200 + 2d


8

10

x

Roy’s Frequent Flyer Miles

f(d)

8

8

–10 –8 –6 –4 –2 –2

19.

y

Frequent Flyer Miles

–8 –10

13,400 13,360 13,320 13,280 13,240 13,200 20 40 60 80 100



d

Amount Charged to Credit Card ($)

20. $900; Sample explanation: Substitute 15,000 for f(d) and then solve for d.




Lesson 2-2 Graphing Systems of Inequalities


PLAN

My Notes

Learning Targets:

• Represent constraints by equations or inequalities. • Use a graph to determine solutions of a system of inequalities.

Pacing: 1 class period Chunking the Lesson #1 #2 #3 #4–5 #6 Check Your Understanding #11 #12–13 #14–15 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Think-Pair-Share, Interactive Word Wall, Create Representations, Work Backward, Discussion Groups, Close Reading, Debriefing, Activating Prior Knowledge

Work with your group on Items 1 through 5. As needed, refer to the Glossary to review translations of key . Incorporate your understanding into group discussions to confirm your knowledge and use of key mathematical language. 1. Roy’s spending money depends on both the number of tickets t and the number of meals m. Determine whether each option is feasible for Roy and provide a rationale in the table below. Tickets (t)

Meals (m)

Total Cost

Is it feasible?

Rationale

6

16

1240

Yes

1240 ≤ 1360

8

14

1360

Yes

1360 = 1360

10

12

1480

No

1480 > 1360

No

You cannot buy half a ticket.

4.5

11

890

2. Construct viable arguments. For all the ordered pairs (t, m) that are feasible options, explain why each statement below must be true. a. All coordinates in the ordered pairs are integer values.



Sample explanation: Both meals and tickets are integral values, so only coordinate pairs that are integers will be solutions.

b. If graphed in the coordinate plane, all ordered pairs would fall either in the first quadrant or on the positive m-axis. Sample explanation: Tickets must be greater than or equal to zero, and meals must be greater than zero.

3. Write a linear inequality that represents all ordered pairs (t, m) that are feasible options for Roy. 100t + 40m ≤ 1360 for t ≥ 0 and m > 0

4. If Roy buys exactly two meals each day, determine the total number of tickets that he could purchase in five days. Show your work. 9 tickets. Sample answer: 100t + 40(10) ≤ 1360 100t ≤ 960 t ≤ 9.6 Because you cannot purchase part of a ticket, at most 9 tickets can be purchased.

Lesson 2-2

#7

TEACH ACADEMIC VOCABULARY The term feasible means that something is possible in a given situation.

DISCUSSION GROUP TIPS As you share your ideas, be sure to use mathematical and academic vocabulary precisely. Make notes as you listen to group to help you the meaning of new words and how they are used to describe mathematical concepts. Ask and answer questions clearly to aid comprehension and to ensure understanding of all group ’ ideas.

MATH A linear inequality is an inequality that can be written in one of these forms, where A and B are not both equal to 0: Ax + By < C, Ax + By > C, Ax + By ≤ C, or Ax + By ≥ C.


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Bell-Ringer Activity Write the statement “You must be at least 13 years old and at least 54 inches tall to ride this ride.” on the board. Write the general ordered pair (age, height) on the board. Ask students which of the following people could ride the ride: Anna(11, 48); Ben(13, 58); Candice(14, 41); Danielle(10, 55); Ed(15, 60). Have students discuss which constraint prevents students from riding the ride. 1 Interactive Word Wall Introduce students to the word feasible as it applies to the solution sets of equations and inequalities. When using algebraic expressions to find solution sets within an applied setting, students must always interpret the solution set for reasonableness within the context of the applied setting. To further illustrate the word feasible, pose the following to students: You and a friend are playing a game of Tic-Tac-Toe. What are the feasible results of the game? [you win, you lose, you tie] 2 Think-Pair-Share This item returns to the domain constraints of the problem. Discuss how those constraints affect the feasible options for the conditions given in Item 1. The statement in Item 2b assumes that Roy will buy some number of meals greater than zero. 3 Create Representations, Debriefing For this item, assume that Roy has no additional expenses. Students may incorrectly assume that everyone eats three meals per day. Therefore, they may be confused by m as a variable for the number of meals that Roy may eat. From their perspective, Roy will eat three meals per day, or 15 meals, during his entire stay in New York City. 4–5 Think-Pair-Share, Work Backward, Discussion Groups In Item 4, some students may give t ≤ 9.6 as the answer. Although this is a correct solution to the inequality, it does not answer the question asked, which requires an integer answer.


21






21 meals. Sample answer: 100(5) + 40m ≤ 1360 40m ≤ 860 m ≤ 21.5 Because you cannot purchase part of a meal, at most 21 meals can be purchased.

Recall how to graph linear inequalities. First, graph the corresponding linear equation. Then choose a test point not on the line to determine which half-plane contains the set of solutions to the inequality. Finally, shade the half-plane that contains the solution set.

6. To see what the feasible options are, you can use a visual display of the values on a graph. a. Attend to precision. Graph your inequality from Item 3 on the grid below. m

Number of Meals

6 Create Representations, Activating Prior Knowledge, Quickwrite Review how to graph a linear inequality. Later, in Item 8, students will graph all the constraint inequalities on one grid. For now, the focus is on one inequality. This makes it easier to determine whether or not students have any misunderstandings about the procedure for graphing linear inequalities. If students need additional help, assign Mini-Lesson: Graphing Linear Inequalities.

MATH TIP

36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16

t

Number of Tickets

b. What is the boundary line of the graph? 100t + 40m = 1360

c. Which half-plane is shaded? How did you decide?

The lower half-plane is shaded. Sample explanation: The test point (0, 0) satisfies the inequality.

22 SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions

MINI-LESSON: Graphing Linear Inequalities If students need additional help with graphing linear inequalities, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.

22



ELL students whose first language is not English by pairing them with more-fluent speakers for Item 6. Pairs can practice their listening and speaking skills as they take turns describing the process of graphing a linear inequality. Encourage students to use precise mathematical language in their discussions.

5. If Roy buys exactly one ticket each day, find the maximum number of meals that he could eat in the five days. Show your work.


4–5 (continued) Likewise, in Item 5, some students may give m ≤ 21.5 as the answer. Although this is a solution to the inequality, the question asks for a specific number of meals, which also must be an integer.

017-028_SB_AG2_SE_U01_A02.indd Page 23 23/01/14 7:28 PM ehi-6





d. Write your response for each item as points in the form (t, m). Item 4 Item 5 (9, 10)

(5, 21)

e. Are both those points in the shaded region of your graph? Explain.

Yes. Sample explanation: Both are near but below the boundary line.

7. Use appropriate tools strategically. Now follow these steps to graph the inequality on a graphing calculator. a. Replace t with x, and replace m with y. Then solve the inequality for y. Enter this inequality into your graphing calculator. 100t + 40m ≤ 1360 →

100x + 40y ≤ 1360 40y ≤ 1360 − 100x y ≤ 34 − 2.5x

TECHNOLOGY TIP To enter an equation in a graphing calculator, start with Y= .

b. Use the left arrow key to move the cursor to the far left of the equation you entered. Press ENTER until the symbol to the left of Y1 changes to . What does this symbol indicate about the graph?

Sample answer: This symbol indicates that the half-plane below the boundary line will be shaded.



c. Now press GRAPH . Depending on your window settings, you may or may not be able to see the boundary line. Press WINDOW and adjust the viewing window so that it matches the graph from Item 6. Then press GRAPH again. Check students’ graphs.

d. Describe the graph.

Sample answer: The graph shows a line with negative slope, with shading below and to the left of the line.

TECHNOLOGY TIP To graph an inequality that includes ≥ or ≤, you would use the symbol or . You need to indicate whether the half-plane above or below the boundary line will be shaded.


Technology Tip If students are using TI-Nspire technology, provide the following directions for how to graph the inequality given in Item 7: Step 1: Choose Graphs&Geometry from the home screen. Step 2: Change the equal sign to a less than sign by using the CLEAR key followed by the < key located in the leftmost column of white keys. Step 3: Enter the function f1(x) as 34 − 2.5 X (use the green letter key for x). Step 4: Adjust the viewing window as needed to view the graph. For additional technology resources, visit SpringBoard Digital. 7 Activating Prior Knowledge, Interactive Word Wall, Debriefing Part a provides an opportunity to review the concept of independent and dependent variables. Discuss that t is replaced with x because t (the number of tickets) is the independent variable and m (the number of meals) depends on the number of tickets. Students should discuss why replacing t with y and m with x does not work. In part c, students can adjust the viewing window by pressing WINDOW and entering an Xmin of 0, an Xmax of 16, an Xscl of 1, a Ymin of 0, a Ymax of 36, and a Yscl of 1. Note that due to the height of the graph on the student page, the graph displayed on the calculator will not completely match, but students should be able to relate one to the other.

23


23




My Notes

Check Your Understanding 8. Compare and contrast the two graphs of the linear inequality: the one you made using paper and pencil and the one on your graphing calculator. Describe an advantage of each graph compared to the other.

Answers

9. a. Solutions on the boundary line represent solutions for which Roy would have no money left over. For these solutions, the total cost of the tickets plus the total cost of the meals is equal to $1360, which is the amount of money Roy has. b. Solutions below the boundary line represent solutions for which Roy would have money left over. For these solutions, the total cost of the tickets plus the total cost of the meals is less than $1360, which is the amount of money Roy has.

MATH TIP Use a solid boundary line for inequalities that include ≥ or ≤. Use a dashed boundary line for inequalities that include > or <.

MATH Constraints are the conditions or inequalities that limit a situation.

10. Explain how you would graph the inequality 2x + 3y < 12, either by using paper and pencil or by using a graphing calculator.

11. Roy realized that some other conditions or constraints apply. Write an inequality for each constraint described below. a. Roy eats lunch and dinner the first day. On the remaining four days, Roy eats at least one meal each day, but he never eats more than three meals each day. 6 ≤ m ≤ 14

b. There are only 10 performances playing that Roy actually wants to see while he is in New York City, but he may not be able to attend all of them. 0 ≤ t ≤ 10

c. Roy wants the number of meals that he eats to be no more than twice the number of performances that he attends. m ≤ 2t

10. Sample answer: Write the related linear equation. Then choose several values of x and substitute them into the equation to find the corresponding values of y. Graph the ordered pairs and draw a dashed line through them. Then choose a test point not on the boundary line and use it to determine which half-plane to shade. 11 Activating Prior Knowledge, Create Representations Students must work with the language of inequalities in such phrases as at least, at most, not more than, less than, more than, and so on. Ask students to distinguish between not more than three and less than three, and to distinguish between at least one and more than one. For each pair of phrases, ask students to use a number line to identify numbers that satisfy both phrases and then numbers that satisfy just one phrase but not the other. Have students represent each phrase by writing an inequality.

9. a. What part of your graphs represents solutions for which Roy would have no money left over? Explain. b. What part of your graphs represents solutions for which Roy would have money left over? Explain.

12. Model with mathematics. You can use a graph to organize all the constraints on Roy’s trip to New York City. a. List the inequalities you found in Items 3 and 11. 6 ≤ m ≤ 14, 0 ≤ t ≤ 10, m ≤ 2t, 100t + 40m ≤ 1360

b. Graph the inequalities from Items 3 and 11 on a single grid.

m 1 • Equations, Inequalities, Functions Answers 24 SpringBoard® Mathematics Algebra 2, Unit

12. b. The shaded region is the intersection of the four inequalities.

26 24 22 20 18 16 14 12 10 8 6 4 2 2 4 6 8 10 12 14 16 Number of Tickets

24


t


8. Sample answer: Both graphs show the same inequality, but the paper graph uses the variables t and m, and the calculator graph uses the variables x and y. One advantage of the paper graph is that you can add titles to the axes to show what the graph represents. One advantage of the calculator graph is that you do not need to determine the coordinates of points on the boundary line in order to graph it.


Debrief students’ answers to these items to ensure that they understand concepts related to graphing linear inequalities and interpreting solutions to situations involving linear inequalities.



Number of Meals






13. By looking at your graph, identify two ordered pairs that are feasible options to all of the inequalities. Confirm that these ordered pairs satisfy the inequalities listed in Item 12. Answers will vary based on the points that students pick.

My Notes

ACADEMIC VOCABULARY When you confirm a statement, you show that it is true or correct.

a. First ordered pair (t, m): b. Second ordered pair (t, m): 14. Label the point (6, 10) on the grid in Item 6. a. Interpret the meaning of this point.

Roy will go to 6 performances and have 10 meals.

b. Construct viable arguments. Is this ordered pair in the solution region common to all of the inequalities? Explain.

14–15 Quickwrite, Discussion Groups, Debriefing Have students list four other points that are solutions and four points that are not solutions. In each case, have them identify why the point is a solution or why it is not. Tie in the graphic representation to the real-world context, and for points that are not solutions, have students write a few sentences explaining why the point is not a solution.

Yes. Sample explanation: The ordered pair is in the shaded region of the graphs of all four inequalities.

15. If Roy uses his prize money to purchase 6 tickets and eat 10 meals, how much money will he have left over for other expenses? Show your work.


$360 Sample answer: 1360 − (6 × 100 + 10 × 40) = 360


12–13 Create Representations As students graph the various inequalities, be sure that they use the boundary lines and shading correctly to interpret the inequalities. Have several students plot the four inequalities on the board so that any differences can be discussed and reconciled. For Item 13, some students may choose points that are not actually solutions of the system of inequalities. Use this as an opportunity to discuss why graphically represented information is only as accurate as the care taken when graphing, and as the precision that a graphing tool allows. Reinforce what it means to be a member of the feasible region and also the importance of verification by substituting into all four inequalities.



Debrief students’ answers to these items to ensure that they understand concepts related to graphing linear inequalities and interpreting solutions to situations involving linear inequalities.

16. Given the set of constraints described earlier, how many tickets could Roy purchase if he buys 12 meals? Explain. 17. a. If you were Roy, how many meals and how many tickets would you buy during the 5-day trip? b. Explain why you made the choices you did, and tell how you know that this combination of meals and tickets is feasible.

Answers 16. 6, 7, or 8 tickets; Sample explanation: Substituting 12 for m and solving m ≤ 2t for t shows that t ≥ 6, so the number of tickets is at least 6. Substituting 12 for m and solving 100t + 40m ≤ 1360 for t shows that t ≤ 8.8, so the number of tickets is no more than 8.

18. Explain how you would graph this constraint on a coordinate plane: 2 ≤ x ≤ 5.


25

17. a. Sample answer: 14 meals and 7 tickets b. Sample answer: If I were Roy, I would want to eat 3 meals per day whenever possible, so I would choose the maximum number of meals, which is 14. If Roy eats 14 meals, he can buy either 7 or 8 tickets and still meet all of the constraints. If I were Roy, I would pick 7 tickets so that I could have some money left over for souvenirs. 18. Graph the vertical lines x = 2 and x = 5. Then shade the region between them.


25




ACTIVITY 2

ASSESS

continued

Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning.

My Notes

19. Graph these inequalities on the same grid, and shade the solution region that is common to all of the inequalities: y ≥ 2, x ≤ 8, and y ≤ 2 + 1 x . 2 20. Identify two ordered pairs that satisfy the constraints in Item 19 and two ordered pairs that do not satisfy the constraints.

See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity.

A snack company plans to package a mixture of almonds and peanuts. The table shows information about these types of nuts. The company wants the nuts in each package to have at least 60 grams of protein and to cost no more than $4. Use this information for Items 21–23.

LESSON 2-2 PRACTICE

19.

10

LESSON 2-2 PRACTICE

y

8

Nut

Protein (g/oz)

Cost ($/oz)

Almonds

6

0.30

Peanuts

8

0.20

6 4 2 –10 –8 –6 –4 –2 –2

2

4

6

8

10

x

–4

MATH TIP

–6 –8 –10

20. Sample answer: ordered pairs that satisfy the constraints: (4, 3) and (6, 2); ordered pairs that do not satisfy the constraints: (5, 1) and (9, 4) 21. 6x + 8y ≥ 60, 0.30x + 0.20y ≤ 4, x ≥ 0, y ≥ 0

22. Graph the constraints. Shade the solution region that is common to all of the inequalities. 23. a. Identify two ordered pairs that satisfy the constraints. b. Reason quantitatively. Which ordered pair represents the more expensive mixture? Which ordered pair represents the mixture with more protein? Explain your answer.

Nut Mixture

y


22.

When answering Item 21, that the number of ounces of each type of nut cannot be negative.

21. Model with mathematics. Write inequalities that model the constraints in this situation. Let x represent the number of ounces of almonds in each package and y represent the number of ounces of peanuts.

16 12 8 4 4

8

12

16

20

d


Ounces of Peanuts

20

Ounces of Almonds

ADAPT Check students’ answers to the Lesson Practice to ensure that they understand how to graph inequalities and identify the solutions of a system of inequalities. Watch for students who choose points that satisfy either inequality as opposed to all inequalities in a system. Reinforce students’ understanding of the meaning of the solution to a system of inequalities.

26a. Answers SpringBoard Mathematics Algebra Inequalities, Functions 23. will vary but should be 2, Unit 1 • Equations, ACTIVITY PRACTICE ordered pairs in the solution region. 1. y = − 2 x + 3 3 Sample answer: (10, 2) and (4, 12) 2. y = −x + 2 b. Answers will vary depending on the answer to part a. Sample answer: y 10 (10, 2) represents a mixture that 8 would cost $3.40 and have 76 g of 6 protein. (4, 12) represents a mixture 4 that would cost $3.60 and have 120 g 1 2 of protein. So, (4, 12) represents both –10 –8 –6 –4 –2 2 4 6 the more expensive mixture and the –2 –4 mixture with more protein. 2 ®

–6 –8

–10

26


8

10

x



Graphing to Find Solutions Choices

ACTIVITY 2 PRACTICE


Lesson 2-1 1. Write the equation of the line with y-intercept 3 and a slope of − 2 , and graph it. 3 2. Write the equation of the line that es through the point (−2, 4) and has a slope of −1. Then graph it. 3. Write the equation of the line in standard form that es through the points (2, −3) and (−1, −4). A jeweler is heating a gold bar. It takes 7 joules of heat to raise the temperature of the bar 1°C. The initial temperature of the bar is 25°C. Use this information for Items 4–11. 4. Make a table that shows how many joules of heat would be required to raise the temperature of the gold bar to 26°C, 27°C, 28°C, 29°C, 30°C, and 35°C. 5. Write the equation of a function h(t) that represents the amount of heat in joules required to heat the bar to a temperature of t degrees Celsius. 6. Graph the function. Be sure to label the axes.

8. What is the t-intercept of the graph of the function? What does it represent in this situation?

continued

A college tennis coach needs to purchase tennis balls for the team. A case of 24 cans costs $60, and each can holds 3 balls. Use this information for Items 12–17.

ACTIVITY PRACTICE 3. x − 3y = 11 4.

Temperature of Bar (°C)

Heat Required (joules)

26

7

27

14

28

21

29

28

30

35

35

70

12. Write the equation of a function c(t) that represents the number of cases the coach will need to purchase to have a total of t tennis balls. 13. Graph the function. Be sure to label the axes. 14. What is the slope of the graph? 1 A. 72 B. 18 C. 8 D. 72 15. What does the slope represent in this situation? 16. Are all the values for t on your graph valid in this situation, given that the coach can only buy complete cases of tennis balls? Explain. 17. The coach needs to purchase 600 tennis balls. a. How many cases will the coach need to purchase to have this number of tennis balls? Explain how you determined your answer. b. What is the actual number of tennis balls the coach will have when he buys this number of cases? c. The coach has a budget of $500 to buy tennis balls. Is there enough money to buy the number of cases that the coach needs? Explain. 18. Without graphing the equations, explain how you can tell which one represents the steeper line: y = 5 + 5(x + 4) or y = 2(3x − 2).

9. Explain what the ordered pair (32, 49) represents in this situation. 10. How many joules of heat will be required to heat the gold bar to a temperature of 260°C? Explain how you determined your answer.

5. h(t) = 7(t − 25) or equivalent 6.

h(t) 84 77 70 63 56 49 42 35 28 21 14 7 24

c(t)

19.

y

Activity 210• Graphing to Find Solutions 8 6 4 2 –10 –8 –6 –4 –2 –2 –4

2

4

6

8

10

x

27

30

32

34

t

Cases of Tennis Balls Needed

10 9 8 7 6 5 4 3 2 1 72

17. a. 9 cases; Sample explanation: I evaluated the function for t = 600 and found that c(600) ≈ 8.3. This value shows that the coach needs more than 8 cases to have 600 tennis balls, so the coach should purchase 9 cases. b. 648 tennis balls c. No. Sample answer: Nine cases will cost $540, which is more than the $500 budgeted for the tennis balls. 18. Sample answer: The slope of y = 5 + 5(x + 4) is 5, and the slope of y = 2(3x − 2) is 6. The line with the greater positive slope is steeper, so y = 2(3x − 2) represents a steeper line than y = 5 + 5(x + 4).

28

7. 7 joules/°C 8. 25; the temperature of the bar before the jeweler begins to heat it 9. Heating the bar to a temperature of 32°C will require 49 joules of heat. 10. 1645 joules; Sample answer: I evaluated the function for t = 260. 11. removing heat from the bar in order to lower its temperature 12. c(t ) = t 72 13.

11. Explain what a negative value of h(t) represents in this situation.

26

Temperature of Bar (˚C)

Cases



7. What is the rate of change of the function, including units?

ACTIVITY 2

Heat Required (joules)


144 216 288 360

t

Tennis Balls

14. A 15. the ratio of the number of cases to the number of tennis balls 16. No. The only values of t that are valid are those that result in a whole number of cases. So, the valid values of t are nonnegative multiples.

–6 –8 –10


27



Graphing to Find Solutions Choices

ACTIVITY 2

y

20. a.

continued

4

2

–2

6

4

8

x

–2

–6 –8

b. Solid line; ≥ means that points on the boundary line are included in the solution set. c. Below the line; (0, 0) is not a solution, so the half-plane that does not include (0, 0) should be shaded. y 12 10

(7, 11)

6 26 5, 5

8 183 , 5 22 11

4 2

–2

2

4

6

8

10 12

x

24. a. Identify two ordered pairs that satisfy the constraints. b. Which ordered pair represents Catelyn working a greater number of hours? Which ordered pair represents Catelyn earning more money? Explain your answer.

Catelyn’s Weekly Work Hours

y 30 25

Nanny Hours

Catelyn has two summer jobs. Each week, she works at least 15 hours at a pet store and at least 6 hours as a nanny. She earns $10 per hour at the pet store and $8 per hour as a nanny. Catelyn wants to work no more than 30 hours and earn at least $250 per week. Use this information for Items 22–25.

23. Graph the constraints. Shade the solution region that is common to all of the inequalities.

22. x ≥ 15, y ≥ 6, x + y ≤ 30, 10x + 8y ≥ 250 23.

21. Graph the following inequalities on the same grid and shade the solution region that is common to all of the inequalities. a. y ≥ 6 − 2 x 3 b. y ≤ 4 + x c. y ≤ 11 − 8(x − 7)

22. Let x represent the number of hours Catelyn works at the pet store in one week and y represent the number of hours she works as a nanny in one week. Write inequalities that model the four constraints in this situation.

6

–6 –4 –2 2

19. Graph the inequality y > 2x − 5. 20. a. Graph the inequality 6x − 2y ≥ 12. b. Did you use a solid or dashed line for the boundary line? Explain your choice. c. Did you shade above or below the boundary line? Explain your choice.

–4

21.

Lesson 2-2

A tent designer is working on a new tent. The tent will be made from black fabric, which costs $6 per yard, and green fabric, which costs $4 per yard. The designer will need at least 3 yards of black fabric, at least 4 yards of green fabric, and at least 10 yards of fabric overall. The total cost of the fabric used for the tent can be no more than $60. Use this information for Items 26–28. 26. Let x represent the number of yards of black fabric and y represent the number of yards of green fabric. Write inequalities that model the four constraints in this situation. 27. Graph the constraints. Shade the solution region that is common to all of the inequalities. 28. Which ordered pair lies in the solution region that is common to all of the inequalities? A. (2, 12) B. (4, 7) C. (6, 8) D. (10, 3)

MATHEMATICAL PRACTICES

Reason Abstractly and Quantitatively Look back at the scenario involving the tent designer. 29. a. What does the ordered pair you chose in Item 28 represent in the situation? b. What is the greatest amount of green fabric the designer can use if all of the constraints are met? Explain your answer. c. What is the least amount of black fabric the designer can use if all of the constraints are met? Explain. © 2015 College Board. All rights reserved.

–4

25. a. Identify two ordered pairs that do not satisfy the constraints. b. For each ordered pair, identify the constraint or constraints that it fails to meet.

20 15 10 5 5

10

15

20

25

30

x

Pet Store Hours


28

27.


y

28. B 29. a. using 4 yards of black fabric and 7 yards of green fabric for the tent b. 10.5 yd; Sample explanation: The point in the shaded region of the graph with the greatest y-value is (3, 10.5). c. 3 yd; Sample explanation: The points in the shaded region of the graph with the least x-value lie on the line x = 3.

12 10

Green Fabric (yd)

24. a. Sample answer: (15, 13) and (20, 10) b. Sample answer: (20, 10); (15, 13) represents working 28 hours and earning $254. (20, 10) represents working 30 hours and earning $280. 25. a. Sample answer: (15, 8) and (17, 15) b. Sample answer: (15, 8) results in her earning only $214. (17, 15) results in her working 32 hours. 26. x ≥ 3, y ≥ 4, x + y ≥ 10, 6x + 4y ≤ 60

8 6 4 2 2

4

6

8

10

12

x

Black Fabric (yd)

28



2

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ACTIVITY

Systems of Linear Equations

ACTIVITY 3

Monetary Systems Overload Lesson 3-1 Solving Systems of Two Equations in Two Variables Learning Targets:

graphing, substitution, and elimination to solve systems of linear • Use equations in two variables. systems of linear equations in two variables to model • Formulate real-world situations.

Activity Standards Focus

My Notes

Have you ever noticed that when an item is popular and many people want to buy it, the price goes up, but items that no one wants are marked down to a lower price?

Suppose that during a six-month time period, the supply and demand for gasoline has been tracked and approximated by these functions, where Q represents millions of barrels of gasoline and P represents price per gallon in dollars. • Demand function: P = −0.7Q + 9.7 • Supply function: P = 1.5Q − 10.4 To find the best balance between market price and quantity of gasoline supplied, find a solution of a system of two linear equations. The demand and supply functions for gasoline are graphed below. p


10 Price (dollars)


12

Lesson 3-1 PLAN CONNECT TO ECONOMICS The role of the desire for and availability of a good in determining price was described by Muslim scholars as early as the fourteenth century. The phrase supply and demand was first used by eighteenth-century Scottish economists.

A point, or set of points, is a solution of a system of equations in two variables when the coordinates of the points make both equations true.

6 4 2 10

Pacing: 2 class periods Chunking the Lesson #1–2 #3 Check Your Understanding #7 Example A #11 Example B Check Your Understanding Lesson Practice

TEACH

MATH

8

5

Guided In Activity 3, students write and graph systems of equations. They solve the systems of equations using graphing, substitution, and elimination. They also use technology and matrices to solve systems of equations. Throughout this activity, emphasize that there is more than one way to solve a system of equations and that some methods are more efficient in certain situations.

SUGGESTED LEARNING STRATEGIES: Shared Reading, Close Reading, Create Representations, Discussion Groups, Role Play, ThinkPair-Share, Quickwrite, Note Taking, Look for a Pattern

The change in an item’s price and the quantity available to buy are the basis of the concept of supply and demand in economics. Demand refers to the quantity that people are willing to buy at a particular price. Supply refers to the quantity that the manufacturer is willing to produce at a particular price. The final price that the customer sees is a result of both supply and demand.

3

Q

Bell-Ringer Activity Have students list five solutions to the equation 2x + y = 14. Then pose and discuss the following questions: 1. Will all students have the same five solutions? 2. How many solutions exist for the equation? 3. How can you visually show all of the existing solutions for the equation?

Developing Math Language Be sure students understand that a solution to a system of equations is any ordered pair that, when substituted into each equation in the system, results in a true statement for every one of the equations in the system. If an ordered pair makes one equation true, but not all of the equations in the system, it is not a solution.

Gasoline (millions of barrels)

1. Make use of structure. Find an approximation of the coordinates of the intersection of the supply and demand functions. Explain what the point represents. Sample answer: (9.15, 3.3); At a price of $3.30, people will demand 9.15 million gallons of gas, and companies will be willing to supply it.

Activity 3 • Systems of Linear Equations

29

Common Core State Standards for Activity 3 HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

1–2 Shared Reading, Close Reading, Interactive Word Wall, Create Representations These first few items introduce solving systems of linear equations by graphing. Item 1 also demonstrates the limitations of graphing as a solution method. It asks students to approximate the solution by identifying a point of intersection that is not a lattice point in the coordinate plane. Review with students that a lattice point is a corner or intersection of two grid lines on the Cartesian plane.


29


ACTIVITY 3 Continued 3 Create Representations Remind students that to graph an equation, they should either write the equation in slope-intercept form or find the x- and y-intercepts.

Lesson 3-1 Solving Systems of Two Equations in Two Variables


2. What problem(s) can arise when solving a system of equations by graphing? Sample answer: Graphing is not very accurate if the intersection is not on a lattice point, or the scaling of the graph is not accurate enough.

Technology Tip Students can use graphing calculators to graph each system and determine its solution. On TI calculators, the intersect option is found as option 5 under the 2nd CALC menu. On a TI-Nspire, this is done under the analyze option in the Graphs&Geometry tool.


TECHNOLOGY TIP You can use a graphing calculator and its Calculate function to solve systems of equations in two variables.

3. Model with mathematics. For parts a–c, graph each system. Determine the number of solutions. y y = x +1 a.   y = −x + 4 6 one solution

4 2


5

–5

x

–2

Developing Math Language Make sure that students understand that although there are four used when describing the solution set for a system of equations, there are only three classifications for a solution set: (1) inconsistent, (2) consistent and independent, (3) consistent and dependent.

 y = 5 + 2 x b.   y = 2 x

y 6

no solutions

4 2 5

–5 –2

 y = 2 x + 1 c.  2 y = 2 + 4 x

4 2 5

–5 –2

d. Graphing two linear equations illustrates the relationships of the lines. Classify the systems in parts a–c as consistent and independent, consistent and dependent, or inconsistent. a. consistent and independent b. inconsistent c. consistent and dependent

30

x


MINI-LESSON: Solving Systems Using a Graphing Calculator If students need additional help solving systems of equations using a graphing calculator, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.



6

infinitely many solutions

Systems of linear equations are classified by the number of solutions. • Systems with one or many solutions are consistent. • Systems with no solution are inconsistent. • A system with exactly one solution is independent. • A system with infinite solutions is dependent.

y


MATH

30

x






Check Your Understanding 4. Describe how you can tell whether a system of two equations is independent and consistent by looking at its graph.

Answers

5. The graph of a system of two equations is a pair of parallel lines. Classify this system. Explain your reasoning. 6. Make sense of problems. A system of two linear equations is dependent and consistent. Describe the graph of the system and explain its meaning.

7. Marlon is buying a used car. The dealership offers him two payment plans, as shown in the table. Payment Plans Plan

Down Payment ($)

Monthly Payment ($)

1

0

300

2

3600

200

CONNECT TO PERSONAL FINANCE A down payment is an initial payment that a customer makes when buying an expensive item, such as a house or car. The rest of the cost is usually paid in monthly installments.



Marlon wants to answer this question: How many months will it take for him to have paid the same amount using either plan? Work with your group on parts a through f and determine the answer to Marlon’s question. a. Write an equation that models the amount y Marlon will pay to the dealership after x months if he chooses Plan 1. y = 300x

b. Write an equation that models the amount y Marlon will pay to the dealership after x months if he chooses Plan 2. y = 3600 + 200x

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to classifying a system of equations by the number of its solutions. To reinforce Item 5, have students make a sketch of the situation.

DISCUSSION GROUP TIP As you work with your group, review the problem scenario carefully and explore together the information provided and how to use it to create a potential solution. Discuss your understanding of the problem and ask peers or your teacher to clarify any areas that are not clear.

4. If the system is independent and consistent, the graph will show a pair of lines that intersect at a point. 5. The system is inconsistent. A pair of parallel lines never intersect, which means that the graphs of the equations have no points in common and the system has no solutions. 6. The graph of the system is a single line; there are an infinite number of solutions. 7 Predict and Confirm, Discussion Groups, Look for a Pattern Prior to using analytic geometry to solve this item, focus student attention on the starting amounts for both plans as well as the rate of change for both s. Students may note that they begin $3,600 apart and that the gap will narrow by $100 each month. Therefore, it will take 36 months for the s to be equal. Connect the initial amounts to the y-intercept and the rates of change to the slopes when solving using analytic geometry.

c. Write the equations as a system of equations. x {yy == 300 3600 + 200 x


31

Common Core State Standards for Activity 3 (continued) HSN-VM.C.6(+)) HSN-VM.C.6(

Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

HSN-VM.C.7(+)

Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

HSN-VM.C.8(+)

Add, subtract, and multiply matrices of appropriate dimensions.

HSN-VM.C.9(+)

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.


31




ACTIVITY 3

Debrief students’ answers to these items to ensure that they understand concepts related to solving a system of equations by graphing. To reinforce Item 9, have students share their answers with a partner and discuss why they chose their answers.

d. Graph the system of equations on the coordinate grid.

14400

When graphing a system of linear equations that represents a real-world situation, it is a good practice to label each line with what it represents. In this case, you can label the lines Plan 1 and Plan 2.

Answers

Used Car Payment Plans

y

MATH TIP

12600 10800 9000

Plan 2

7200

Plan 1

5400 3600 1800

8. Sample answer: Check that the ordered pair (36, 10,800) satisfies both of the equations in the system. 9. Plan 1; The graph shows that when the time is less than 36 months (or 3 years), the total amount paid for Plan 1 is less than the total amount paid for Plan 2. 10. Sample answer: Identify the two quantities in the situation that can vary. Assign variables to these quantities. Write an equation in of the two variables that models part of the situation. Then write a second equation in of the two variables that models another part of the situation. Finally, write the two equations as a system.

6

12

18 24 30 Time (months)

36

42

x

e. Reason quantitatively. What is the solution of the system of equations? What does the solution represent in this situation? (36, 10,800); In 36 months, the total cost of both plans will be $10,800.

f. In how many months will the total costs of the two plans be equal? 36 months

Check Your Understanding 8. How could you check that you solved the system of equations in Item 7 correctly? 9. If Marlon plans to keep the used car less than 3 years, which of the payment plans should he choose? Justify your answer. 10. Construct viable arguments. Explain how to write a system of two equations that models a real-world situation.

32


Common Core State Standards for Activity 3 (continued) HSN-VM.C.10(+)) Understand that the zero and identity matrices play a role in matrix addition and HSN-VM.C.10( multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

32

HSA-REI.C.8(+)

Represent a system of linear equations as a single matrix equation in a vector variable.

HSA-REI.C.9(+)

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).




My Notes


Tables of values can be used to answer Item 7. Creating and populating the tables of values often helps students who struggle with algebraic modeling to write equations correctly.

Total Amount Paid ($)


continued






Investors try to control the level of risk in their portfolios by diversifying their investments. You can solve some investment problems by writing and solving systems of equations. One algebraic method for solving a system of linear equations is called substitution.

My Notes

TEACHER to TEACHER

Example A During one year, Sara invested $5000 into two separate funds, one earning 2 percent and another earning 5 percent annual interest. The interest Sara earned was $205. How much money did she invest in each fund? Step 1: Let x = money in the first fund and y = money in the second fund. Write one equation to represent the amount of money invested. Write another equation to represent the interest earned. x + y = 5000 The money invested is $5000. 0.02x + 0.05y = 205 The interest earned is $205. Step 2: Use substitution to solve this system. x + y = 5000 Solve the first equation for y. y = 5000 − x 0.02x + 0.05(5000 − x) = 205 Substitute for y in the second equation.



Step 3:

Example A Note Taking Walk students through the example. Some students may find it easier to work with whole numbers. Have them multiply the second equation by 100 to rewrite it as 2x + 5y = 20,500.

MATH In the substitution method, you solve one equation for one variable in of another. Then substitute that expression into the other equation to form a new equation with only one variable. Solve that equation. Substitute the solution into one of the two original equations to find the value of the other variable.

Students may struggle with this example because they fail to understand the problem. Have students solve the problem first through guess-and-check. This guess-and-check process will ensure that they understand the problem and will also motivate students to learn a more efficient way to find the solution.

0.02x + 250 − 0.05x = 205 Solve for x. −0.03x = −45 x = 1500 Substitute the value of x into one of the original equations to find y.

x + y = 5000 1500 + y = 5000 Substitute 1,500 for x. y = 3500 Solution: Sara invested $1500 in the first fund and $3500 in the second fund.

Try These A

MATH TIP Check your answer by substituting the solution (1500, 3500) into the second original equation, 0.02x + 0.05y = 205

Write your answers on notebook paper. Show your work. Solve each system of equations, using substitution.

Check students’ work.

 x = 25 − 3 y a.   4 x + 5 y = 9 (−14, 13)

 x + 2 y = 14 b.  2 y = x − 10 (12, 1)

 y − x = 4 c.  3x + y = 16 (3, 7)

d. Model with mathematics. Eli invested a total of $2000 in two stocks. One stock cost $18.50 per share, and the other cost $10.40 per share. Eli bought a total of 130 shares. Write and solve a system of equations to find how many shares of each stock Eli bought.

{18x +.5yx=+130 10.4 y = 2000 (80, 50); 80 shares at $18.50, 50 shares at $10.40


33


33




TEACHER to TEACHER Students may question why they have to learn more than one way to solve a system of equations. Allow students to compare and contrast the methods by having them solve one or more of the following systems using each method. 1. 7x + 5y = −1 4x − y = −16 [(−3, 4)] 2. 3x − 2y = −21 2x + 5y = 5 [(−5, 3)]

My Notes

11. When using substitution, how do you decide which variable to isolate and which equation to solve? Explain. Sample answer: Choose a variable that is easy to isolate by finding the equation with a variable that has a coefficient of 1 or −1.

Another algebraic method for solving systems of linear equations is the elimination method.

Example B MATH In the elimination method, you eliminate one variable. Multiply each equation by a number so that the for one variable combine to 0 when the equations are added. Then use substitution with that value of the variable to find the value of the other variable. The ordered pair is the solution of the system. The elimination method is also called the addition-elimination method or the linear combination method for solving a system of linear equations.

A stack of 20 coins contains only nickels and quarters and has a total value of $4. How many of each coin are in the stack? Step 1: Let n = number of nickels and q = number of quarters. Write one equation to represent the number of coins in the stack. Write another equation to represent the total value. n + q = 20 5n + 25q = 400 Step 2:

To solve this system of equations, first eliminate the n variable. −5(n + q) = −5(20) 5n + 25q = 400 −5n − 5q = −100 5n + 25q = 400

Step 3:

3. 7x + 5y = 9 4x − 3y = 11 [(2, −1)] Step 4:

The number of coins is 20. The total value is 400 cents. Multiply the first equation by −5.

Add the two equations to eliminate n.

20q = 300 Solve for q. q = 15 Find the value of the eliminated variable n by using the original first equation. n + q = 20 n + 15 = 20 Substitute 15 for q. n=5 Check your answers by substituting into the original second equation.

5n + 25q = 400 5(5) + 25(15) = 400 Substitute 5 for n and 15 for q. 25 + 375 = 400 400 = 400 Solution: There are 5 nickels and 15 quarters in the stack of coins.

34

34




Example B Note Taking Work through the example with students. Refer to the Math box for a summary of how to use the elimination method. Point out the importance of multiplying both sides of one equation by a number that will allow one variable term to be eliminated when the equations are added.




11 Think-Pair-Share, Look for a Pattern Have volunteers share their answers to this item. Focus a discussion on why it is helpful to look for a variable with a coefficient of 1 first, and then, if there are no such variables, to look for a variable with a coefficient of −1 next.





Try These B Solve each system of equations using elimination. Show your work. Check students’ work.

−2 x − 3 y = 5 a.  −5x + 3 y = −40 (5, −5)

5x + 6 y = −14 b.   x − 2 y = 10 (2, −4)

Answers

−3x + 3 y = 21 c.  −x − 5 y = −17

12. Sample answer: In both methods, you start by solving for the value of one of the variables and then use that value to solve for the value of the other variable. In the substitution method, you use substitution to get rid of one of the variables. In the elimination method, you add equations to get rid of one of the variables. 13. Sample answer: The first equation; to solve the first equation for x, Ty only needs to add 2y to both sides, but to solve the second equation for x, Ty would need to do two steps: first, subtract 6y from both sides, and then divide both sides by 4. 14. Sample answer: Multiply the second 2 x − 4 y = 15 equation by 2 to get  . 6 x + 4 y = 18 Then add the equations to eliminate the variable y and get 8x = 33.

(−3, 4)

d. A karate school offers a package of 12 group lessons and 2 private lessons for $110. It also offers a package of 10 group lessons and 3 private lessons for $125. Write and solve a system of equations to find the cost of a single group lesson and a single private lesson.

{1210gg ++ 23pp == 110 125

; (5, 25); $5 for a group lesson, $25 for a private lesson

Check Your Understanding 12. Compare and contrast solving systems of equations by using substitution and by using elimination. x − 2 y = 8 using 13. Reason abstractly. Ty is solving the system   4 x + 6 y = 10 substitution. He will start by solving one of the equations for x. Which equation should he choose? Explain your reasoning. 14. Explain how you would eliminate one of the variables in this


2 x − 4 y = 15 system:  . 3x + 2 y = 9


Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to solving a system of equations by substitution and by elimination.


LESSON 3-1 PRACTICE 15. Solve the system by graphing.

2 x + 9 = y  y = −4 x − 3 

16. Solve the system using substitution.

 4 y + 19 = x 3 y − x = −13 

17. Solve the system using elimination.

3x + 2 y = 17 4 x − 2 y = 4 


18. Make sense of problems and persevere in solving them. At one company, a level I engineer receives a salary of $56,000, and a level II engineer receives a salary of $68,000. The company has 8 level I engineers. Next year, it can afford to pay $472,000 for their salaries. Write and solve a system of equations to find how many of the engineers the company can afford to promote to level II.

TEACHER to TEACHER In Item 18, let one variable represent the number of engineers who will stay at level I, and let the other variable represent the number of engineers who will be promoted to level II.

19. Which method did you use to solve the system of equations in Item 18? Explain why you chose this method. Activity 3 • Systems of Linear Equations

LESSON 3-1 PRACTICE

15. (−2, 5) 16. (−5, −6) 17. (3, 4)  x + y = 8 18.  , 56, 000 x + 68, 000 y = 472, 000 where x is the number of engineers who will stay at level I and y is the number of engineers who will be promoted to level II; solution: (6, 2); The company can afford to promote 2 engineers to level II. 19. Answers will vary.

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ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing systems of equations and solving systems of equations by graphing, substitution, and elimination. If students are having difficulty writing equations that model a situation, review the steps of identifying what you know and what you want to know, asg variable names and writing equations based on what you know.


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ACTIVITY 3 Continued Lesson 3-2

Lesson 3-2 Solving Systems of Three Equations in Three Variables


PLAN My Notes

Pacing: 1 class period Chunking the Lesson Example A Example B #1 #2–5 #6–7 Check Your Understanding Lesson Practice

systems of three linear equations in three variables using • Solve substitution and Gaussian elimination. systems of three linear equations in three variables to model a • Formulate real-world situation. SUGGESTED LEARNING STRATEGIES: Close Reading, Vocabulary Organizer, Note Taking, Summarizing, Paraphrasing, Graphic Organizer, Group Presentation, Think Aloud, Identify a Subtask

TEACH

Sometimes a situation has more than two pieces of information. For these more complex problems, you may need to solve equations that contain three variables.

Bell-Ringer Activity Write the following on the board: If z = 15, find the ordered pair (x, y) that satisfies the system of equations: x + y − z = −1 2x − 2y + 3z = 8

In Bisbee, Arizona, an old mining town, you can buy souvenir nuggets of gold, silver, and bronze. For $20, you can buy any of these mixtures of nuggets: 14 gold, 20 silver, and 24 bronze; 20 gold, 15 silver, and 19 bronze; or 30 gold, 5 silver, and 13 bronze. What is the monetary value of each souvenir nugget?

Discuss students’ answers and have them share how they solved the problem.

The problem above represents a system of linear equations in three variables. The system can be represented with these equations. 14 g + 20s + 24b = 20 20 g + 15s + 19b = 20   30 g + 5s + 13b = 20

Introduction Close Reading, Vocabulary Organizer Solving this contextual problem is not part of this activity. The problem is used simply to illustrate one type of situation that can be represented by three variables in a system of linear equations.

Although it is possible to solve systems of equations in three variables by graphing, it can be difficult. Just as the ordered pair (x, y) is a solution of a system in two variables, the ordered triple (x, y, z) is a solution of a system in three variables. Ordered triples are graphed in three-dimensional coordinate space.


(3, –2, 4) 4 units up

O 2 units left

x

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3 units forward



y


z


The point (3, −2, 4) is graphed below.

Many students have trouble visualizing a three-dimensional coordinate system when it is represented with a twodimensional drawing. Help students understand the three-dimensional coordinate system by using the physical classroom as a model. The floor of the classroom can be thought of as Quadrant I of the xy-plane and one corner of the room can be thought of as the positive z-axis. Ading rooms on the same floor can be visualized to extend the xy-plane, and rooms below your classroom can be visualized to represent the negative z-axis.

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Learning Targets:






You can use the substitution method to solve systems of equations in three variables.

My Notes

Example A

2 x + 7 y + z = −53  Solve this system using substitution. −2 x + 3 y + z = −13  6 x + 3 y + z = −45 Step 1:

Step 2:


Solve the first equation for z.

For students who need a challenge beyond Example 3, assign the problem of finding the monetary value of the souvenir nuggets on the previous page. The correct solution is g = $0.50, s = $0.35, b = $0.25.

2x + 7y + z = −53 z = −2x − 7y − 53 Substitute the expression for z into the second equation. Then solve for y.

−2x + 3y + z = −13 −2x + 3y + (−2x − 7y − 53) = −13 Substitute −2x − 7y − 53 for z. −4x − 4y − 53 = −13 Solve for y. −4y = 4x + 40 y = −x − 10 Step 3:

Example A Note Taking Point out that solving systems in three variables is similar to solving systems in two variables once a variable term has been eliminated. In this case, after the x- are eliminated from the three original equations, students will be solving a system in two variables—y and z.

Use substitution to solve the third equation for x.

6x + 3y + z = −45 6x + 3y + (−2x − 7y − 53) = −45 Substitute −2x − 7y −53 for z. 4x − 4y − 53 = −45 4x − 4y = 8 4x − 4(−x − 10) = 8 Substitute −x − 10 for y. 4x + 4x + 40 = 8 Solve for x. 8x = −32 x = −4



Step 4:

Solve the last equation from Step 2 for y. y = −x − 10 y = −(−4) − 10 y = −6

Step 5:

MATH TIP

Substitute −4 for x.

Solve the last equation from Step 1 for z.

z = −2x − 7y − 53 z = −2(−4) − 7(−6) − 53 Substitute −4 for x and −6 for y. z = 8 + 42 − 53 z = −3 Solution: The solution of the system is (−4, −6, −3).

As a final step, check your ordered triple solution in one of the original equations to be sure that your solution is correct.


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ACTIVITY 3 Continued Example B Work Backward Students often have trouble locating the source of the error when they do not arrive at the correct solution when solving a system of equations in three variables. Encourage them to write notes about what they are doing and label equations so that they can more easily retrace their steps when checking their work.



Solve each system of equations using substitution. Show your work.  x + 4 y + z = 3 3x + y + z = 5   b.  x + 2 y − 3z = 15 a. 2 x + y + z = 11    4 x + y + 2z = 23 2 x − y + z = 2 (5, −1, 2)

Universal Access For students who are struggling with solving systems in three variables in which all three variables are present in all three equations, consider beginning with the following two systems. 1. x + y − z = 4; y + z = 12; y = 6 [(4, 6, 6)] 2. x + y + z = 10; x − y = −1; x + y = 5 [(2, 3, 5)]

Try These A

MATH When using Gaussian elimination to solve a system of three equations in the variables x, y, and z, you start by eliminating x from the second and third equations. Then eliminate y from the third equation. The third equation now has a single variable, z; solve the third equation for z. Then use the value of z to solve the second equation for y. Finally, use the values of y and z to solve the first equation for x.

(3, 0, −4)

Another method of solving a system of three equations in three variables is called Gaussian elimination. This method has two main parts. The first part involves eliminating variables from the equations in the system. The second part involves solving for the variables one at a time.

2 x + y − z = 4  Solve this system using Gaussian elimination. −2 x + y + 2z = 6   x + 2 y + z = 11

Example B

Step 1: Use the first equation to eliminate x from the second equation. 2x + y − z = 4 −2x + y + 2z = 6 2y + z = 10  2 x + y − z = 4  2 y + z = 10    x + 2 y + z = 11

Add the first and second equations.

Replace the second equation in the system with 2y + z = 10.

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Multiply the third equation by −2.

2x + y −z = 4 −2x −4y −2z = −22 −3y −3z = −18

Add the equations to eliminate x.

2 x + y − z = 4  2 y + z = 10    −3 y − 3z = −18

Replace the third equation in the system with −3y − 3z = −18.




2x + y − z = 4 −2(x + 2y + z) = −2(11)


Step 2: Use the first equation to eliminate x from the third equation.





Step 3:


Use the second equation to eliminate y from the third equation.

Challenge students to write a system of three equations in three variables that has the solution (3, −1, 5). Point out that there are an infinite number of systems with this solution. If necessary, provide the hint that students need to work backward.

My Notes

3(2y + z) = 3(10) Multiply the second equation by 2(−3y − 3z) = 2(−18) 3 and the third equation by 2. 6y + 3z = 30 −6y − 6z = −36 −3z = −6 2 x + y − z = 4  2 y + z = 10   −3 z = −6 

Add the equations to eliminate y.

Replace the third equation in the system with −3z = −6.

Step 4:

Solve the third equation for z.

CONNECT TO MATH HISTORY

Step 5:

−3z = −6 z=2 Solve the second equation for y.

Step 6:

2y + z = 10 2y + 2 = 10 Substitute 2 for z. 2y = 8 y=4 Solve the first equation for x.

The method of Gaussian elimination is named for the German mathematician Carl Friedrich Gauss (1777–1855), who used a version of it in his calculations. However, the first known use of Gaussian elimination was a version used in a Chinese work called Nine Chapters of the Mathematical Art, which was written more than 2000 years ago. It shows how to solve a system of linear equations involving the volume of grain yielded from sheaves of rice.

2x + y − z = 4 2x + 4 − 2 = 4 Substitute 4 for y and 2 for z. 2x + 2 = 4 2x = 2 x=1 Solution: The solution of the system is (1, 4, 2).


Try These B



a. Solve this system of equations using Gaussian elimination. Show your work.

1 Think-Pair-Share Have students trade flowcharts with a partner and then follow that flowchart to solve the system 2x − y + z = 10; 3x − 2y − 2z = 7; x − 3y − 2z = 10. Students can critique their partner’s flowcharts based on their effectiveness for solving the given system.

TEACHER to TEACHER No two students seem to follow the exact same steps when solving a system of equations in three variables, which can make it difficult when students ask you for help. Have students verbally walk you through what they have done so far rather than beginning the solution process from scratch with them. It is important to model to students that understanding others’ mathematical thinking is a good practice.

 2 x + y − z = −2   x + 2 y + z = 11 (−2, 5, 3)  −2 x + y + 2z = 15

1. Make a flowchart on notebook paper that summarizes the steps for solving a system of three equations in three variables by using either substitution or Gaussian elimination. Check students’ flowcharts.


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39




ACTIVITY 3

Tell students that an equation in three variables represents a plane in three-dimensional space. Have students draw diagrams showing the possible ways to position three planes in three-dimensional space and then label the diagrams with the consistent, inconsistent, dependent, and independent.

My Notes

2. The farmer has 500 acres to plant with corn, soybeans, and wheat. Write an equation in of c, s, and w that models this information. c + s + w = 500

2–5 Create Representations, Guess and Check, Sharing and Responding After students have written their equations, have them make guesses for the values of the three variables. Students should discuss their strategies for making educated rather than random guesses.

3. Growing an acre of corn costs $390, an acre of soybeans costs $190, and an acre of wheat costs $170. The farmer has a budget of $119,000 to spend on growing the crops. Write an equation in of c, s, and w that models this information. 390c + 190s + 170w = 119,000

6–7 Critique Reasoning, Discussion Groups Ask students to consider and discuss the following questions:

4. The farmer plans to grow twice as many acres of wheat as acres of corn. Write an equation in of c and w that models this information.

1. Why are three equations necessary for a word problem that contains three variables? 2. Would the solution to the problem still be (150, 50, 300) if the constraint from Item 4 were not included as part of the problem?

2c = w

 c + s + w = 500   390c + 190s + 170w = 119, 000     2c = w

Determine the reasonableness of your solution. Does your answer make sense in the context of the problem?

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6. Make sense of problems. Solve the system of equations. Write the solution as an ordered triple of the form (c, s, w). Explain the meaning of the numbers in the ordered triple. (150, 50, 300); The farmer should grow 150 acres of corn, 50 acres of soybeans, and 300 acres of wheat. This will meet his budget and the requirement to grow twice as many acres of wheat as corn.




5. Write your equations from Items 3–5 as a system of equations.

MATH TIP

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A farmer plans to grow corn, soybeans, and wheat on his farm. Let c represent the number of acres planted with corn, s represent the number of acres planted with soybeans, and w represent the number of acres planted with wheat.



continued






7. Explain what the solution you found in Item 6 represents in the real-world situation.

The farmer should plant 150 acres of corn, 50 acres of soybeans, and 300 acres of wheat.

Answers 8. Sample answer: For consistent and independent systems, the solution of a system of two linear equations in two variables is an ordered pair, and the solution of a system of three linear equations in three variables is an ordered triple. Both types of systems can be solved by using substitution or a type of elimination. 9. Sample answer: To eliminate x from the second equation, add the first and second equations in the system. To eliminate x from the third equation, start by multiplying the first equation by −2. Then add the resulting equation to the third equation.

Check Your Understanding 8. Compare and contrast systems of two linear equations in two variables with systems of three linear equations in three variables. 9. Explain how you could use the first equation in this system to eliminate x from the second and third equations in the system:  x + 2 y − z = 5 −x − y + 2z = −13.    2 x + y − 2z = 14

LESSON 3-2 PRACTICE

 x − 3 y + z = −15  2 x + y − z = −2    x + y + 2z = 1 3x + y − z = 4  11. Solve the system using Gaussian elimination. −3x + 2 y + 2z = 6   x − y + 2z = 8 10. Solve the system using substitution.


Use the table for Items 12–14.



Frozen Yogurt Sales Time Period

Small Cups Sold

Medium Cups Sold

Large Cups Sold

1:00–2:00

6

10

8

97.60

2:00–3:00

9

12

5

100.80

3:00–4:00

10

12

4

99.20

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to solving a system of equations with three variables.


Sales ($)

LESSON 3-2 PRACTICE

12. Write a system of equations that can be used to determine s, m, and l, the cost in dollars of small, medium, and large cups of frozen yogurt. 13. Solve your equation and explain what the solution means in the context of the situation. 14. Use appropriate tools strategically. Which method did you use to solve the system? Explain why you used this method.


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10. (−3, 4, 0) 11. (2, 2, 4) 6s + 10m + 8l = 97.60  12. 9s + 12m + 5l = 100.80  10s + 12m + 4l = 99.20 13. (3.20, 4.00, 4.80); A small cup costs $3.20, a medium cup costs $4.00, and a large cup costs $4.80. 14. Sample answer: I used Gaussian elimination because if I had used substitution instead, I would have ended up with equations that had fractional coefficients.

ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing systems of equations and solving systems of equations using Gaussian elimination and substitution. If students are having difficulty using Gaussian elimination, review strategies for reordering the equations and deciding which equation to use to eliminate variables from the other equations.


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Lesson 3-3 Matrix Operations

ACTIVITY 3

Lesson 3-3

continued

PLAN

WRITING MATH You can name a matrix by using a capital letter.

TEACH

• Add, subtract, and multiply matrices. • Use a graphing calculator to perform operations on matrices.

SUGGESTED LEARNING STRATEGIES: Interactive Word Wall, Note Taking, Close Reading, Summarizing, Paraphrasing, Discussion Groups, Work Backward

A matrix, such as matrix A below, is a rectangular arrangement of numbers written inside brackets.  2 4 5  A= −3 8 −2 

READING MATH

Bell-Ringer Activity Provide students with the tables below and have them determine the gross sales for the day. [$533.11] Smalls Sold

Mediums Sold

Larges Sold

23

45

71

Cost for Small

$2.49

Cost for Medium

$3.49

Cost for Large

$4.49

1–3 Close Reading, Interactive Word Wall, Graphic Organizer Students often confuse the column and row and reverse the order of the numbers in both the dimensions and the addresses of entries of a matrix. Point out that an architectural column is vertical, as is a column in a matrix.

To say the dimensions of matrix A, read 2 × 3 as “2 by 3.”

MATH A matrix (plural: matrices) is a rectangular array of numbers arranged in rows and columns inside brackets. The dimensions of a matrix are the number of rows and the number of columns, indicated by m × n, where m is the number of rows and n is the number of columns. The entries of a matrix are the numbers in the matrix.

The dimensions of a matrix give its number of rows and number of columns. A matrix with m rows and n columns has dimensions m × n. Matrix A has 2 rows and 3 columns, so its dimensions are 2 × 3. The numbers in a matrix are called entries. The address of an entry gives its location in the matrix. To write the address of an entry, write the lowercase letter used to name the matrix, and then write the row number and column number of the entry as subscripts. The address a12 indicates the entry in matrix A in the first row and second column, so a12 is 4. In the next lesson, you will learn how to use matrices to solve systems of equations. Use these matrices to answer Items 1–3.  3 −6    B =  8 10    0  −4

14 40   C=  26 30 

1. What are the dimensions of each matrix? matrix B: 3 × 2; matrix C: 2 × 2

2. Make use of structure. Write the entry indicated by each address. b. b12 −6 a. b31 −4 d. c22 30 c. c21 26

Developing Math Language Have students practice by writing the table for Items 12–14 in Lesson 3-2 in matrix form. They should be careful to distinguish the row and column labels from the entries in the matrix. Have students determine the dimensions of the matrix.

3. What is the address of the entry 8 in matrix B? Explain. b21; The entry 8 is in the second row and the first column.

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#1–3 #4 Example A Check Your Understanding Example B Check Your Understanding Lesson Practice

Learning Targets:


My Notes

Pacing: 1-2 class periods Chunking the Lesson





You can input a matrix into a graphing calculator using the steps below. Step 1: Go to the Matrix menu. To do this, press 2nd , and then press the key with MATRIX printed above it. Step 2: Use the right arrow key to select the Edit submenu. Step 3: Move the cursor next to the name of one of the matrices and press ENTER to select it. Step 4: Enter the correct dimensions for the matrix.


My Notes

TECHNOLOGY TIP When entering a negative number as an entry in a matrix, be sure to use the negative key to enter the negative sign, not the subtraction key.

Step 5: Enter the entries of the matrix. To save the matrix, press 2nd , and then press the key with QUIT printed above it. 4. Input each matrix into a graphing calculator. Check students’ work.  6 −1  a. A =  4   5

The word matrix has many definitions that are unrelated to mathematics. Some students may be able to make a better language connection with the word array. For students who may also not understand the word corresponding, provide a visual cue to help them understand the phrase corresponding entries.

If two matrices have the same dimensions, you can add or subtract them by adding or subtracting their corresponding entries.

Example A  2 8 10   C= −3 1 −5

TECHNOLOGY TIP  5 0 −2   D=  8 7 −4 



a. Find C + D.   C +D =  2+5 −3 + 8

  10 + (−2)  =  7 −5 + (−4)  5

8+0 1+ 7

8 8

 8  −9 

Example A Graphic Organizer, Note Taking Begin a graphic organizer of matrix operations with students. Include rules for determining if the operation is defined, a description of how to carry out the operation, and a method for determining the dimensions of the answer matrix.

ELL

 3 −2 5  b. B =  0 −4   6

Find each matrix sum or difference.

Technology Tip

To add the matrices on a graphing calculator, first input both matrices. Then select [C] from the Names submenu of the Matrix menu. Press + . Then select [D] from the Names submenu of the Matrix menu. Your screen should now show [C]+[D]. Press ENTER to show the sum.

b. Find C − D.  2 − 5 8 − 0 10 − (−2)  −3 8 12  =  C−D =  −3 − 8 1 − 7 −5 − (−4) −11 −6 −1

Try These A Find each matrix sum or difference.  6 −2   E= 4   9 a. E + F  5 3   19 11

 −1 5   F=  10 7  b. E − F  7 −7    −1 −3 

c. F − E −7 7     1 3 


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My Notes


Answers 5. Sample answer: A table is a way of arranging information in rows and columns, and a matrix is a way of arranging numbers in rows and columns. 6. Matrix addition is commutative.  −2 4   and Sample example: A =   0 3  1 2  B= −3 4  ;  −1 6   A+B = B+A =  −3 7  7. The matrices do not have the same dimensions. 2  −7 8.   ; Sample explanation: I  0 −4  wrote the opposite of each entry in the original matrix.

5. How is a matrix similar to a table?

MATH TIP Recall that the Commutative Property of Addition states that a + b = b + a for any real numbers a and b.

 4 −2    B =  5 −6     6 −10 

8. Two matrices are additive inverses if each entry in their sum is 0. What is the additive inverse of the matrix shown below? Explain how you determined your answer.  7 −2    4   0

You can also find the product of two matrices A and B if the number of columns in A is equal to the number of rows in B. For example, the dimensions of matrix A below are 3 × 2, and the dimensions of matrix B are 2 × 1. The matrix product AB is defined because A has 2 columns and B has 2 rows.  2 8   A =  −7 5    1 3  

MATH TIP The inner dimensions of two matrices indicate whether their product is defined. n×m

m×p

The outer dimensions indicate the dimensions of the matrix product. n×m

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7. Explain why you cannot subtract these two matrices.  2 3 4  A= −1 −6 −8 

Differentiating Instruction If students need a challenge, ask them to devise a way to change a matrix subtraction problem to a matrix addition problem by “adding the opposite.” Ask students to share their methods with the class.

6. Express regularity in repeated reasoning. Make a conjecture about whether matrix addition is commutative. Then provide an example that s your conjecture.

 4 B=  −2 

The product of an n × m matrix and an m × p matrix is an n × p matrix. Because A above is a 3 × 2 matrix and B is a 2 × 1 matrix, the product AB is a 3 × 1 matrix. To find the entry in row i and column j of the product AB, find the sum of the products of consecutive entries in row i of matrix A and column j of matrix B. To see what this means, take a look at the next example.

m×p




Debrief students’ answers to these items to ensure that they understand concepts related to matrices and matrix operations.










Example B

Find the matrix product AB. 1 5 −4   A= 2   3 −2

2 4   B =  3 −1    0 −2 

TEACHER to TEACHER

Step 1:

Determine whether AB is defined.

Step 2:

A is a 2 × 3 matrix, and B is a 3 × 2 matrix, so AB is defined. A has 2 rows and B has 2 columns, so AB is a 2 × 2 matrix. Find the entry in row 1, column 1 of AB. Use row 1 of A and column 1 of B. Multiply the first entries, the second entries, and the third entries. Then add the products. 1(2) + 5(3) + (−4)(0) = 17 2 4  1  5 −4   17 _    3 −1 =   AB =  2    3 −2   _ _   0 −2 

Step 3:

Find the entry in row 1, column 2 of AB. Use row 1 of A and column 2 of B. 1(4) + 5(−1) + (−4)(−2) = 7

Step 4:

Example B Close Reading Emphasize the importance of determining the dimensions of the product matrix prior to performing the operation. Identifying the address of each entry in the product matrix helps students to focus on the correct row and column to use in the factor matrices.

TECHNOLOGY TIP To multiply the matrices on a graphing calculator, first input both matrices. Then select [A] from the Names submenu of the Matrix menu. Press x . Then select [B] from the Names submenu of the Matrix menu. Your screen should now show [A]*[B]. Press ENTER to show the product.

2 4  1  5 −4   17 7    3 −1 =   AB =  2    3 −2   _ _   0 −2  Find the entry in row 2, column 1 of AB.

Use this visual technique to help students struggling with matrix multiplication. When finding the  3 1   product AB, where A = −2 4    0 −3  1 −5   , write the problem and B =  3 −4 in stacked form. Each entry in the answer matrix is an intersection of a row and a column in the factor matrices: 1 -5 -4 3 3 -2 0

1 4 -3

P31

P22



Use row 2 of A and column 1 of B.

Technology Tip

3(2) + (−2)(3) + 2(0) = 0 2 4  1  5 −4   17 7    3 −1 =   AB =  2    3 −2   0 _   0 −2  Step 5:


Find the entry in row 2, column 2 of AB. Use row 2 of A and column 2 of B. 3(4) + (−2)(−1) + 2(−2) = 10

2 4  1  5 −4   17 7    3 −1 =   AB =  2    3 −2   0 10   0 −2  17 7   Solution: AB =   0 10 


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My Notes

Find each matrix product if it is defined.

4  3

 1 4  C= −7 6 

12   12 

 −8  −80

c. ED 21 −37   −11 −47 

not defined

Check Your Understanding 9. Is matrix multiplication commutative? Provide an example that s your answer. 10. The matrix product RS is a 3 × 4 matrix. If R is a 3 × 2 matrix, what are the dimensions of S? Explain your answer. 11. Critique the reasoning of others. Rebekah made an error when finding the matrix product KL. Her work is shown below. What mistake did Rebekah make? What is the correct matrix product?  2 1 8 5   K= L= −4 −2   0 −3  2(1) 8(5)  2 40  =  KL =  −4(0) −2(−3)  0 6 


LESSON 3-3 PRACTICE Use these matrices to answer Items 12–17.  3 −6 1  A= 2 0  −8

 5 −7   B= 2  −3

 4 10   C= −1 −3

12. What are the dimensions of A? 13. Look for and make use of structure. What is the entry with the address b12?

LESSON 3-3 PRACTICE

12. 2 × 3 13. −7

14. Find B + C. 15. Find C − B. 16. Find AB if it is defined. 17. Find BC if it is defined.

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Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to Matrix operations. If students are having difficulty multiplying matrices, have them recite with you, “To find the entry in the first row and first column, multiply the first row times the first column,” while you circle the first row of the first factor and the first column of the second factor.

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 8 5 −1  E= −4 4 −9 

b. CE

−16 −12    −24 −86 

ASSESS

ADAPT

 0 8  D= −4 −5

a. CD

−2 10   and BA =   6 0  10. 2 × 4; The number of rows of S must equal the number of columns of R, so S has 2 rows. The number of columns of S must equal the number of columns of RS, so S has 4 columns. 11. Rebekah incorrectly found the entries of KL by multiplying the corresponding entries of K and L. Instead, she should have found each entry klij in KL by finding the sum of the products of consecutive entries in row i of matrix K and row j of matrix L.  2 −14   KL =  −4 −14 

 9 3  14. B + C =  −4 −1 −1 17   15. C − B =   2 −5 16. not defined  27 71  17. BC =  −14 −36 

Try These B



 −2 9. No. Sample example: A =   0  1 2 −14  ; AB =  and B =  −3 4   −9

continued


Debrief students’ answers to these items to ensure that they understand concepts related to matrix products.

Answers


ACTIVITY 3





Lesson 3-4 Solving Matrix Equations


Lesson 3-4 PLAN

Learning Targets:

systems of two linear equations in two variables by using graphing • Solve calculators with matrices. systems of three linear equations in three variables by using • Solve graphing calculators with matrices.

My Notes

Pacing: 1-2 class periods Chunking the Lesson Check Your Understanding #5–7 Example A #8 #9–10 #11–12 #13–14 #15 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Vocabulary Organizer, Note Taking, Discussion Groups, Marking the Text, Debriefing, Identify a Subtask, Create Representations, Look for a Pattern A square matrix is a matrix with the same number of rows and columns. A multiplicative identity matrix is a square matrix in which all entries along the main diagonal are 1 and all other entries are 0. The main diagonal of a square matrix is the diagonal from the upper left to the lower right. A multiplicative identity matrix is often called an identity matrix and is usually named I. A 3 × 3 identity matrix is shown below. The entries in blue are on the main diagonal. 1 0 0   I = 0 1 0    0 0 1  The product of a square matrix and its multiplicative inverse matrix is an identity matrix I. The multiplicative inverse of matrix A is often called the inverse of A and may be named as A−1. So, by definition, A−1 is the inverse of A if A • A−1 = I.

MATH A square matrix is a matrix in which the number of rows equals the number of columns.

TEACH

A multiplicative identity matrix, or identity matrix, is a square matrix in which all entries are 0 except the entries along the main diagonal, all of which are 1.

Write matrices A and B (shown below) on the board and have students find AB and BA. [I3×3]

An identity matrix is the product of a square matrix and its multiplicative inverse matrix, or inverse matrix.

MATH TIP



Check Your Understanding 0 1 1. Construct viable arguments. Explain why   is not an identity matrix.  1 0 

Not all square matrices have inverses. But if a square matrix has an inverse, it must be a square matrix.

Bell-Ringer Activity

 0. 2 3 0 0. 2 0  2     0. 3 1  A =  2 0 −2  ; B =  − 0. 2     1  0.2 −0.3 0   0 1

Developing Math Language Use the Bell-Ringer Activity to emphasize the square matrix, multiplicative identity matrix, and multiplicative inverse matrix. Identify A, B, and the products AB and BA as square matrices. Identify AB and BA as multiplicative identity matrices. Identify B as the multiplicative inverse matrix of A and B as the multiplicative inverse matrix of A.


Use these matrices and a graphing calculator to answer Items 2–4. 2 4  1. 5 −2    A= B= 1  1 3  −0.5

Debrief students’ answers to these items to ensure that they understand concepts related to identity and inverse matrices.

Answers

2. Multiply A by a 2 × 2 identity matrix. Describe the relationship between the matrix product AI and A. 3. Is B the inverse of A? Explain. 4. Is A the inverse of B? Explain.

You can use matrices and inverse matrices to solve systems of linear equations.


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1. In an identity matrix, all entries on the main diagonal are 1, and all the rest of the entries are 0. In this matrix, all entries on the main diagonal are 0, and all the rest of the entries are 1. 2 4  ; The matrix product AI 2. AI =   1 3 is the same as A. 3. Yes; the matrix product AB is the  1 0 identity matrix  .  0 1 4. Yes; the matrix product BA is the  1 0 identity matrix  .  0 1


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5–7 Activating Prior Knowledge, Create Representations The notation A−1 that is used to denote the multiplicative inverse of a matrix A can seem strange to students. Remind them that the multiplicative inverse of the number 8 is 1 , which is equal to 8−1. 8 Point out, however, that A−1 does not mean that students should take the multiplicative inverse of each entry in the matrix.

Technology Tip For additional technology resources, visit SpringBoard Digital.

My Notes

MATH TIP The second equation in the system, x − 4y = −2, can be written as 1x + (−4)y = −2, which more clearly shows that the coefficient of x is 1 and the coefficient of y is −4.

The first step in solving a system of linear equations by using matrices is to write the system as a matrix equation. The diagram shows how to write the 2 x + 3 y = 7 as a matrix equation. system   x − 4 y = −2 A •X= B 2 3  x   7     =    1 −4   y  −2  coefficient matrix A

MATH A matrix equation is an equation of the form AX = B. • A is the coefficient matrix, the matrix formed by the coefficients of the system of equations. • X is the variable matrix, a column matrix that represents all the variables of the system of equations. • B is the constant matrix, a column matrix representing all the constants of the systems of equations.

TECHNOLOGY TIP If the entries in the inverse matrix are decimals, try converting them to fractions by pressing MATH and selecting 1: Frac.

constant matrix B

variable matrix X To solve a matrix equation AX = B for X, you use a process similar to what you would use when solving the regular equation ax = b for x. To solve ax = b, you could multiply both sides of the equation by the multiplicative inverse of a. Likewise, to solve the matrix equation AX = B, you can multiply both sides of the equation by the multiplicative inverse matrix of A. Thus, the solution of AX = B is X = A−1B. You can use a graphing calculator to help you find A−1. 2 3  x   7    =  . In Items 5–7, use the matrix equation   1 −4   y  −2  5. To solve for X, you first need to find A−1. Input matrix A, the coefficient matrix, into a graphing calculator. Then select [A] from the Names submenu of the Matrix menu. Then press x−1 . Your screen should now show [A]−1. Press ENTER to show the inverse matrix. What is A−1? 4 3  11 A−1 =  11 1 −2  11 11

6. Find the matrix product A−1B.  44 33   11 22 −11 11  77 = 11 =   A− B= = 11 A B −22 11  11 − − 22  − 11 11 11 11

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Be certain that students can properly identify the coefficients, variables, and constant in the equation 2x − 4y = 12 prior to having them learn the coefficient matrix, variable matrix, and constant matrix.









7. Make sense of problems. What are the values of x and y in the variable matrix X? How do you know? x

My Notes

x = 2; y = 1; X is equal to A−1B. Because X =   and  y  2 A−1B =   , x must equal 2 and y must equal 1.  1

Example A Note Taking Walk students through the example. Have half of the class use the matrix equation as written and the other half reverse the order of the system of equations in the matrix equation. Ask students to predict how this change might affect the variable matrix found in Step 3. Try These A

Answers  2 1  x   8     =   ; (5, −2) a.   5 6   y  13

Example A The hourly cost to a police department of using a canine team depends on the hourly cost x in dollars of using a dog and the hourly salary y of a handler. The hourly cost for a team of three dogs and two handlers is $82, and the hourly cost for a team of four dogs and four handlers is $160. The 3x + 2 y = 82 system  4 x + 4 y = 160 models this situation. Use a matrix equation to  solve the system, and explain what the solution means. Step 1: Use the system to write a matrix equation.  3 2   x   82    =   4 4   y  160  Step 2:

Enter the coefficient matrix A and the constant matrix B into a graphing calculator. MATRIX[A] 2 × 2 [3 2 ] [4 4 ]



Step 3:

 6 −3  x  −18  b.    =  ; (1, 8) 4   y   34   2  1 −2   x  −23   =  ; (−3, 10) c.  3  y   21  3

MATH TIP Before you can write a matrix equation to model a system of two linear equations in two variables, each equation in the system must be written in standard form Ax + By = C, where A, B, and C are real numbers.

MATRIX[B] 2 × 1 [82 ] [160 ]

Use the calculator to find A−1B.

TEACHER to TEACHER Some students may question whether it is possible to find a multiplicative inverse matrix without using a calculator. Direct these students to research the Augmented Matrix Method and the Adt Method. Research on these methods may certainly make students appreciate their graphing calculators or other technology.

[A]-1 * [B] [ [2] [38] ]

Step 4: Identify and interpret the solution of the system. Solution: The matrix product A−1B is equal to the variable matrix X, so x = 2 and y = 38. The solution of the system is (2, 38). The solution shows that the hourly cost of using a dog is $2 and the hourly salary of a handler is $38.

Try These A Write a matrix equation to model each system. Then use the matrix equation to solve the system. 2 x + y = 8 a.  5x + 6 y = 13

6 x − 3 y = −18 b.  2 x + 4 y = 34

 x − 2 y = −23 c.  3x + 3 y = 21


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ACTIVITY 3 Continued 8 Debriefing As a follow-up to this item, ask students how they would go about finding the exact answers to a system of equations if they found that the calculator was providing them with rounded answers.



8. What is an advantage of using a graphing calculator to solve a system of two linear equations in two variables as opposed to solving the system by making a hand-drawn graph? Sample answer: A solution you obtain from a hand-drawn graph is likely to be an estimate and may not be very accurate. A solution you obtain from a graphing calculator (either from graphing the system or from solving a matrix equation) is likely to be more accurate, although the x- and y-coordinates may be rounded.

9–10 Debriefing, Create Representations Reinforce the structure of the matrix equation by having students manually multiply the coefficient matrix by the variable matrix.

You can also use a matrix equation to solve a system of three linear equations in three variables. Use this information to complete Items 9–15. Karen makes handmade greeting cards and sells them at a local store. The cards come in packs of 4 for $11, 6 for $15, or 10 for $20. Last month, the store sold 16 packs containing 92 of Karen’s cards for a total of $223. The following system models this situation where x is the number of small packs, y is the number of medium packs, and z is the number of large packs.  x + y + z = 16  4 x + 6 y + 10z = 92   11x + 15 y + 20z = 223 9. If you were to model the system with a matrix equation, what would be the dimensions of the coefficient matrix? How do you know?

 1 1 1  x   16        4 6 10   y  =  92  11 15 20   z   223       

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10. Model with mathematics. Write a matrix equation to model the system.


3 × 3; The matrix will have 3 rows because the system has 3 equations. It will have 3 columns because each equation has 3 coefficients.






11. Use appropriate tools strategically. Use a graphing calculator to find A−1, the inverse of the coefficient matrix.

My Notes

11–12 Create Representations, Debriefing Be certain that students know how to toggle between fraction and decimal representation on their calculators. 13–14 Create a Plan, Create Representations, Identify a Subtask These items continue to walk students through the steps for solving the word problem presented prior to Item 9. Point out that students will want to follow these same steps to solve subsequent word problems even though those items no longer reference these steps.

 5 −2  5 6 3   −1  3 1 A = −5 − 2   2 − 1  1  3 3 

12. Use a graphing calculator to find A−1B. 8 A−1B =  5  3  

15 Discussion Groups Have students share their answers within small groups. Each group should prepare a final answer to the question and present it to the class.

13. Find the solution of the system of equations and explain the meaning of the solution. (8, 5, 3); The store sold 8 small packs of Karen’s cards, 5 medium packs, and 3 large packs.



14. How can you check that you found the solution of the system correctly? Sample answer: I could substitute the solution into one of the original equations of the system to check that it satisfies the equation.

15. Work with your group. Compare and contrast using a matrix equation to solve a system of two linear equations in two variables with using a matrix equation to solve a system of three linear equations in three variables. Sample answer: Both processes are the same, except for the dimensions of the matrices. For a system of two equations, the coefficient matrix is 2 × 2, the variable matrix is 2 × 1, and the constant matrix is 2 × 1. For a system of three equations, the coefficient matrix is 3 × 3, the variable matrix is 3 × 1, and the constant matrix is 3 × 1.

DISCUSSION GROUP TIP As you listen to the group discussion, take notes to aid comprehension and to help you describe your own ideas to others in your group. Ask questions to clarify ideas and to gain further understanding of key concepts.


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ACTIVITY 3

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to solving systems of equations with matrices. Ensure students are using appropriate and precise mathematical terminology.

continued My Notes

Check Your Understanding 16. A system of equations and the matrix equation that models it are shown below. Find AX, the product of the coefficient matrix and the variable matrix of the matrix equation. What is the relationship between AX and the system of equations?

Answers

 3x + 2 y   ; The entries in AX 16. AX =   2 x + 4 y  are the same as the variable expressions in the system of equations. 17. Doug tried to solve for X by finding the matrix product AB. Instead, he should have found A−1B. 18. Sample answer: I get an error message when I try to find A−1B on a graphing calculator. I think this probably indicates that the system has no solutions. A graph of the system shows two parallel lines, which confirms that the system is inconsistent and has no solutions.

3x + 2 y = 18 2 x + 4 y = 20 

17. Critique the reasoning of others. Doug incorrectly solved the  3 2  18  matrix equation in Item 17 by finding the matrix product    .  2 4   20  What mistake did Doug make? What should he have done instead? 2 x + y = 6 18. What happens when you try to solve the system   4 x + 2 y = 16 by writing and solving a matrix equation? What do you think this result indicates about the system? Confirm your answer by graphing the system and using the graph to classify the system.

LESSON 3-4 PRACTICE

y

2 8  19. Use a graphing calculator to find the inverse of the matrix  .  5 4 

10 8

For Items 20–21, write a matrix equation to model each system. Then use the matrix equation to solve the system.

4x + 2y = 16

2 x + 4 y = 22 20. −3x + 2 y = 7 

4 2 –6

–4

2

–2

4

6

x

–2


CONNECT TO ECONOMICS The euro is the unit of currency used in many of the nations in the European Union. The British pound is the unit of currency used in the United Kingdom.

 x + y + z = 11  21. 2 x + 8 y + 3z = 80   4 x − 6 y + 7 z = −62

22. Model with mathematics. Steve has 2 euros and 4 British pounds worth a total of $9.10. Emily has 3 euros and 1 British pound worth a total of $5.55 a. Write a system of equations to model this situation, where x represents the value of 1 euro in dollars and y represents the value of 1 British pound in dollars. b. Write the system of equations as a matrix equation. c. Use the matrix equation to solve the system. Then interpret the solution. 23. Which solution method for solving systems of equations do you find easiest to use? Which method do you find most difficult to use? Explain why.


ADAPT Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to writing and solving matrix equations. If students are having difficulty setting up the matrix equation, have them check their equation by manually finding the matrix product AX = B. The result should be the original system of equations.

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LESSON 3-4 PRACTICE

− 1 1  8 4   19.  5 − 1  32 16  2 4   x   22     =   ; (1, 5) 20.  −3 2   y   7  1 1 1  x   11      21.  2 8 3  y  =  80  ; (3, 10, −2)       4 −6 7   z  −62 

2 x + 4 y = 9.10 22. a.  3x + y = 5.55  2 4   x   9.10     = b.   3 1  y   5.55 c. (1.31, 1.62); A euro is worth $1.31, and a British pound is worth $1.62. 23. Answers will vary.



6


2x + y = 6

 3 2   x  18     =    2 4   y   20 



Systems of Linear Equations Monetary Systems Overload


ACTIVITY 3 PRACTICE

1. Solve the system by graphing.  y = −3x + 6  y = −1 x −2  3 

2 –10 –8 –6 –4 –2 –2

Ingredient

2 x + 3 y = 7 4. The system of equations  has 10 x + cy = 3 solutions for all values of c except:

Peanut butter Grape jelly

A. −15 C. 10

B. −3 D. 15  y = −2 x − 1 . 5. a. Graph the system  3 y = −3 − 6 x

Granola

b. Classify the system, and tell how many solutions it has. 6. Mariana had a $20 gift card to an online music store. She spent the entire amount on songs, which cost $1 each, and music videos, which cost $2 each. Mariana bought five more songs than music videos. Write and solve a system of equations to find the number of songs and the number of music videos Mariana bought. 7. A chemist needs to mix a 2% acid solution and a 10% acid solution to make 600 milliliters of a 5% acid solution. Write and solve a system of equations to find the volume of the 2% solution and the volume of the 10% solution that the chemist will need.

6

8

10

x

–8 –10

2. 3. 4. 5.

(−40, −9) (29, 10) D a.

Grams of Fat per Ounce

168

14

71

0

4

132

12

2

y 6

An 18-ounce jar of the sandwich spread will have a total of 2273 calories and 150 grams of fat. Write and solve a system of equations to find the number of ounces of peanut butter, grape jelly, and granola in each jar. 11. A small furniture factory makes three types of tables: coffee tables, dining tables, and end tables. The factory needs to make 54 tables each day. The number of dining tables made per day should equal the number of coffee tables and end tables combined. The number of coffee tables made each day should be three more than the number of end tables. Write and solve a system of equations to find the number of tables of each type the factory should make each day. 12. Can you solve this system? Explain.  x + 2 y + 3z = 8 3x + 4 y + 5z = 10 

12. No; Sample explanation: The system includes 3 variables, but there are only 2 equations. To solve a system involving 3 variables, you need at least 3 equations.

4

–6

Calories per Ounce

–6

53

–4

2

–2

4

6

x

–2 –4 –6

6.

7.


2

–4

10. A snack company plans to sell a mixture of peanut butter, grape jelly, and granola as a sandwich spread. The table gives information about each ingredient.

3. Solve the system using elimination. 2 x − 5 y = 8  x − 3 y = −1 


6 4

9. Solve the system using Gaussian elimination. 2 x + 4 y + z = 31 −2 x + 2 y − 3z = −9    x + 3 y + 2z = 21

2. Solve the system using substitution. 5 y − x = −5 7 y − x = −23

y

8

8. Solve the system using substitution.  x + y + z = 6 2 x + y + 2z = 14   3x + 3 y + z = 8

Lesson 3-1


10

Lesson 3-2


 x + y + z = 54 11.  y = x + z , where x is the number   x = z + 3 of coffee tables, y is the number of dining tables, and z is the number of end tables made each day; (15, 27, 12); The factory should make 15 coffee tables, 27 dining tables, and 12 end tables each day.

ACTIVITY PRACTICE 1. (3, −3)

8. 9. 10.

b. The system is consistent and dependent. It has infinitely many solutions.  s + 2m = 20  s − 5 = m , where s is the  number of songs and m is the number of music videos; (10, 5); Mariana bought 10 songs and 5 music videos.  x + y = 600 0.02 x + 0.10 y = 0.05(600) , where  x is the number of milliliters of the 2% solution and y is the number of milliliters of the 10% solution; (375, 225); The chemist should mix 375 milliliters of the 2% solution and 225 milliliters of the 10% solution. (3, −2, 5) (7, 4, 1)  x + y + z = 18 168 x + 71 y + 132z = 2273, where   14 x + 12z = 150 x is the number of ounces of peanut butter, y is the number of ounces of grape jelly, and z is the number of ounces of granola; (9, 7, 2); Each jar of the sandwich spread will contain 9 ounces of peanut butter, 7 ounces of grape jelly, and 2 ounces of granola.


53



21. The product of 2 matrices is defined if the number of columns in the first matrix is equal to the number of rows in the second matrix. The product matrix has the same number of rows as the first matrix and the same number of columns as the second matrix.  1  3  − 1 0 − 2  4 22. a.  b.  4 1 − 3 1  1   2 2  2  23. Yes. The matrix product AB is the  1 0 identity matrix   , which means  0 1 that B is the inverse of A. The  1 0 matrix product BA is also  ,  0 1 which means that A is the inverse of B. −3x − 2 y = 1  x   1 24.  ; =  5x + 4 y = −3  y  −2  3 2 −7   x  −29       25.  4 −6 5  y  = −19  ; (−2, 6, 5)      1 −4   z  −30   8  x + y = 6 26.  , where x is 3.48 x + 2.64 y = 19.62 the number of pounds of beef and y is the number of pounds of pork; 1 1   x  6  ;    =  3.48 2.64   y  19.62  (4.5, 1.5); Guillermo bought 4.5 pounds of beef and 1.5 pounds of pork.  3 4 5  k   87.5  k   2.5          27.  1 1 2   m =  31.5 ;  m =  5  ;           2 0 1  g   17   g  12  Key chains are $2.50, mugs are $5, and gift wrap is $12. 28. Sample answer: To solve both types of equations, you multiply both sides by a multiplicative inverse. To solve ax = b, you multiply both sides by the multiplicative inverse of a, which is the reciprocal of a. The solution of ax = b is x = b . To solve AX = B, a you multiply both sides by the multiplicative inverse matrix of A, which you can determine by using a graphing calculator. The solution of AX = B is X = A−1B.

Systems of Linear Equations Monetary Systems Overload


Lesson 3-3 Use the given matrices for Items 13–20.  3 0 1  A= −1 2 6   3 −1 4   D= 3 1  2

 1 2 3  4 −2     C =  3 2 1 B= 5  1  −2 4 3     4 1  E=  2 3

24. Write the system of equations represented by the matrix equation below. Then solve the matrix equation. −3 −2   x   1    =   4   y  −3  5 25. Write a matrix equation to model the system. Then use the matrix equation to solve the system. 3x + 2 y − 7 z = −29  4 x − 6 y + 5z = −19   8 x + y − 4 z = −30

13. What are the dimensions of matrix A? 14. What is the entry with the address c13? 15. Find A + D. 16. Find B − E. 17. Find ED if it is defined. 18. Find AC if it is defined. 19. Find AB if it is defined. 20. Let P equal the matrix product BA. Which expression gives the value of P12? A. −2(3) + 5(−1) B. 1(3) + 5(1) C. 4(0) + (−2)(2) D. 4(0) + 1(2) 21. Explain how to determine whether the product of two matrices is defined and how to determine the dimensions of a product matrix.

26. Guillermo bought ground beef and ground pork for a party. The beef costs $3.48/lb and the pork costs $2.64/lb. Guillermo bought 6 pounds of meat for a total of $19.62. Write a system of equations that can be used to determine how many pounds of each type of meat Guillermo bought. Then use a matrix equation to solve the system. 27. Dean, John, and Andrew sold key chains, mugs, and gift wrap for a school fundraiser. The table below shows the number of items that each person sold and the amount of money collected from the sales. Write a matrix equation that can be used to find the price for each item in the table. Then solve the equation to find the prices.

22. Find the inverse of each matrix.  3 1 −2 −1   a.  b.  3  2 0   2 23. Are these matrices inverses of each other? Explain. 0 4  −2 1   A= B= 1 8   0.25 0 

Mugs

Gift Wrap

Amount of Sales

Dean

3

4

5

$87.50

John

1

1

2

$31.50

Andrew

2

0

1

$17.00


Look For and Make Use of Structure 28. Compare and contrast solving an equation of the form ax = b for x with solving a matrix equation of the form AX = B for X.

54



54

Key Chains

Lesson 3-4



19. not defined

3  0 −3    2  −1  1 10 12    −7 26 17  20. C


13. 2 × 3 14.  6 −1 5   16. 15.  5 7   1 14 −1 17   18. 17.  7 11 12

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Embedded Assessment 1 Use after Activity 3

GAMING SYSTEMS A gaming manufacturing company is developing a new gaming system. In addition to a game console, the company will also produce an optional accessory called a Jesture that allows s to communicate with the game console by using gestures and voice commands.

Embedded Assessment 1 Assessment Focus • Systems of equations • Systems of inequalities • Absolute value equations

Answer Key 1. a. t = 15g − 50 b.

Solve the following problems about the gaming system. Show your work. 1. The company plans to sell the video game console at a loss in order to increase its sales. It will make up for the loss from profits made from the sales of games for the system. The company will lose $50 for each console it sells and earn a profit of $15 for each game sold for the system. a. Write an equation that can be used to determine t, the total amount the company will earn from a customer who buys a console and g games. b. Graph the equation on a coordinate grid. c. The company predicts that the average customer will buy seven games for the video game console. What is the total amount the company will earn from the average customer who buys a game console and seven games?

100

Total Earnings ($)

80


40 20 0

1

2

3

4

5

6

g

7

–40 –60

Games Purchased

c. $55 2. a. 9x + y ≤ 8500, 9x + 3y ≤ 9000, 4x + 8y ≤ 12,000, x ≥ 0, y ≥ 0

Plant 1 Plant 2 (hours per system) (hours per system) 9 1 9 3 4 8

b. y

Number of Gaming Systems per Week

1600

Gaming Systems from Plant 2


Motherboard production Technical labor General manufacturing

60

–20

2. To produce the new system, the company plans on using resources in two manufacturing plants. The table gives the hours needed for three tasks. For both plants combined, the company has allocated the following resources on a weekly basis: no more than 8500 hours of motherboard production, no more than 9000 hours of technical labor, and no more than 12,000 hours of general manufacturing. Resources

Total Earnings per Customer

t

a. Write inequalities that model the constraints in this situation. Let x represent the number of gaming systems that will be made in Plant 1, and let y represent the number of gaming systems that will be made in Plant 2. b. Graph the constraints. Shade the solution region that is common to all of the inequalities. c. Identify an ordered pair that satisfies the constraints. Explain what the ordered pair represents in the context of the situation. 3. The Jesture accessory can recognize players when they are within a certain range. The player’s distance from the Jesture can vary up to 1.2 meters from the target distance of 2.4 meters. a. Write an absolute value equation that can be used to find the extreme distances that a player can stand from the Jesture and still be recognized. b. Solve your equation, and interpret the solutions.

1400 1200 1000 800 600 400 200 200 400 600 800 1000 1200 1400

x

Gaming Systems from Plant 1


55

Common Core State Standards for Embedded Assessment 1 HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions. HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. HSA-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

c. Sample answer: (600, 900); The company could meet the constraints by making 600 gaming systems per week at Plant 1 and 900 gaming systems per week at Plant 2. 3. a. |d − 2.4| = 1.2 or equivalent, where d represents the extreme distances in meters a player can stand from the Jesture and still be recognized b. d = 3.6, d = 1.2; The greatest distance a player can stand from the Jesture and be recognized is 3.6 meters, and the least distance the player can stand from the Jesture and be recognized is 1.2 meters.


55


or equivalent, where x is the number of fitness points earned per minute of yoga, y is the number of fitness points earned per minute of aerobics, and z is the number of fitness points earned per minute of jogging. b. (1, 2, 4); A player earns 1 point per minute of yoga, 2 points per minute of aerobics and 4 points per minute of jogging.

Use after Activity 3

GAMING SYSTEMS 4. The Jesture will come with a fitness program. The program allows players to earn fitness points depending on the number of minutes they spend on each activity. The table shows how many minutes three players spent on each activity and the total number of fitness points they earned.

Cassie Clint Kian

Aerobics (minutes) 10 20 25

Jogging (minutes) 20 10 20

Fitness Points 130 95 140

a. Write a system of three equations that can be used to determine the number of points a player gets for 1 minute of each activity. b. Solve your system, and interpret the solution.

Use this first Embedded Assessment as an opportunity to focus on your expectations for what student work should look like as the course progresses. Discuss multiple representations, emphasizing that students should show their work on each part of a question when asked.

Mathematics Knowledge and Thinking

TEACHER to TEACHER

Problem Solving

You may wish to read through the scoring guide with students and discuss the differences in the expectations at each level. Check that students understand the used.

Mathematical Modeling / Representations

Scoring Guide

Exemplary

understanding of solving systems of equations and inequalities, and absolute value equations

56

Incomplete

of solving systems of equations and inequalities and absolute value equations

solving systems of equations and inequalities and absolute value equations

• Little or no understanding

of solving systems of equations and inequalities and absolute value equations

• No clear strategy when

• Fluency in representing

• Little difficulty representing • Partial understanding of


•

•

real-world scenarios using linear equations, systems of equations and inequalities, and absolute value equations Clear and accurate understanding of creating graphs of equations and inequalities

• Ease and accuracy in

explaining interpreting solutions in the context of a real-world scenario

SpringBoard® Mathematics Algebra 2


• A functional understanding • Partial understanding of

strategy that results in a correct answer

(Items 1a, 1b, 2a, 2b, 3a, 4a)

(Items 2c, 3b, 4b)

Emerging

• An appropriate and efficient • A strategy that may include • A strategy that results in

(Items 1c, 2c, 3b, 4b)

Reasoning and Communication

Proficient

The solution demonstrates these characteristics:

• Clear and accurate

(Items 2b, 3b, 4b)

Unpacking Embedded Assessment 2

56

Yoga (minutes) 30 15 10

Play Tester

TEACHER to TEACHER

Once students have completed this Embedded Assessment, turn to Embedded Assessment 2 and unpack it with them. Use a graphic organizer to help students understand the concepts they will need to know to be successful on Embedded Assessment 2.



unnecessary steps but results in a correct answer

real-world scenarios using linear equations, systems of equations and inequalities, and absolute value equations Mostly accurate creation of graphs of equations and inequalities

some incorrect answers

•

how to represent real-world scenarios using linear equations, systems of equations and inequalities, and absolute value equations Partially accurate creation of graphs of equations and inequalities

• Little difficulty in explaining • Partially correct interpreting solutions in the context of a real-world scenario

explanations and interpretations of solutions in the context of a real-world scenario

solving problems

•

of how to represent real-world scenarios using linear equations, systems of equations and inequalities, and absolute value equations Inaccurate or incomplete creation of graphs of equations and inequalities

• Incomplete or inaccurate

explanations and interpretations of solutions in the context of a real-world scenario


4. a. 30 x + 10 y + 20z = 130 15x + 20 y + 10z = 95   10 x + 25 y + 20z = 140





ACTIVITY

Piecewise-Defined Functions

ACTIVITY 4

Absolutely Piece-ful Lesson 4-1 Introduction to Piecewise-Defined Functions

• Graph piecewise-defined functions. the domain and range of functions using interval notation, • Write inequalities, and set notation.

Activity Standards Focus

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Quickwrite, Create Representations, Interactive Word Wall, Marking the Text, Think-Pair-Share, Discussion Groups The graphs of both y = x − 2 for x < 3 and y = −2x + 7 for x ≥ 3 are shown on the same coordinate grid below. y

Lesson 4-1

6

PLAN

4

Pacing: 1 class period Chunking the Lesson

2 5

–5

x

#1–2 #3–4 #5 Check Your Understanding #9 #10 Check Your Understanding Lesson Practice

–2 –4 –6

1. Work with your group on this item and on Items 2–4. Describe the graph as completely as possible.

DISCUSSION GROUP TIP

2. Make use of structure. Why is the graph a function?

As you listen to your group’s discussions as you work through Items 1–4, you may hear math or other words that you do not know. Use your math notebook to record words that are frequently used. Ask for clarification of their meaning, and make notes to help you and use those words in your own communications.



Sample answer: The graph shows an increasing linear function with a slope of 1 for x-values from negative infinity to 3; it then shows a decreasing linear function with a slope of −2 for x-values from 3 to infinity.

Sample answer: It is a function because for each x-value, there is only one y-value, and graphically, it es the vertical line test.

3. Graph y = x2 − 3 for x ≤ 0 and y = 1 x + 1 for x > 0 on the same 4 coordinate grid. y 6 4 2 5

–5

x

–2 –4 –6

Activity 4 • Piecewise-Defined Functions

57

Common Core State Standards for Activity 4 HSF-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. HSF-IF.C.7

Guided In Activity 4, students identify and graph various piecewise-defined functions. They explore functions made up of parts of linear functions. They look at the absolute value and step functions. Finally, they transform various parent piecewise functions. Throughout this activity, emphasize the use of technology to graph piecewise-defined functions as well as how changes in coefficients and constants affect the graphs of functions.

My Notes

Learning Targets:

4

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

HSF-IF.C.7b Graph piecewise-defined functions, including step functions and absolute value functions.

TEACH Bell-Ringer Activity Ask students to graph the following and be ready to share a description of the features of each graph. 1. y = −3x + 5 [graph of line, slope of −3, y-intercept of 5] 2. y = x2 − 6 [graph of parabola, opening upward, with y-axis as line of symmetry, and with vertex at (0, −6)] 3. y = x+ 1 [V-shaped with the point of the “V” at (0, 1); note nothing is below the x-axis] Have students discuss the methods they used to graph these functions. 1–2 Activating Prior Knowledge, Create Representations, Quickwrite If students have previous experience with piecewise-defined functions, their descriptions will provide formative information about their understanding. Students are asked to graph two equations on the same grid and recognize the combined graphs as a function. Students may describe the graphs as “pieces” that are connected or “pieces” that are not connected. It is imperative that students recognize the graphs on each grid as functions, as this provides the access to defining piecewise-defined functions.


57



Lesson 4-1 Introduction to Piecewise-Defined Functions


Sample answer: The graph has a break at x = 0. To the left of x = 0, it is a decreasing quadratic function with a minimum at y = −3, which is included. To the right of x = 0, but not including x = 0, it is an increasing linear function with a slope of 1 . The graph is a function because for 4 each x-value, there is only one y-value.

Technology Tip

1. Press the y = key. 2. For Y1, type in (x2 − 2)(x < −3). Note: In order to access the < symbol, press the 2nd key, followed by the MATH key to access the “Test.” 3. For Y2, type in (2x + 1)(−3 ≤ x) (x < 5). Note: You key in the domain of Y2 separately. 4. For Y3, type in (8)(x ≥ 5).

MATH A piecewise-defined function is a function that is defined using different rules for the different nonoverlapping intervals of its domain.

The functions in Items 1 and 3 are piecewise-defined functions . Piecewisedefined functions are written as follows (using the function from Item 3 as an example):

Name of function

Function Rules

5. Model with mathematics. Complete the table of values. Then graph the function. −2 x − 2 g (x ) =   x + 3


MATH TIP A piecewise-defined function is so called because it is a function that follows a different set of rules as the domain changes. Each set of rules provides a different “piece” of the function. The pieces can be straight or curved. Piecewise-defined functions exist because in the real world not every situation can be modeled using only a single function. The rules may change as the domain changes. 5 Create Representations As students complete the table, make sure that they are using the correct rule for the given domain values. Additionally, when graphing, students may stop the graph for the left-hand piece before x = −1 and not continue to show x = −1 with an open circle. If this occurs, ask them to consider the domain of the left part of the graph and then ask them which rule applies to points between −2 and −1.

Domain Restriction

 x 2 − 3 if x ≤ 0 f (x ) =  1  x + 1 if x > 0  4

5. Press the GRAPH key.


Single Brace

A piecewise-defined function may have more than two rules. For example, consider the function below. −2 if x < −1 h( x ) =  x 2 if − 1 ≤ x < 1  if x ≥ 1 1

58

if x < −1 if x ≥ −1 y

x

g(x)

−4

6

−3

4

−2

2

−1

2

0

3

–4

1

4

–6

2

5

6 4 2


MINI-LESSON: Recognizing and Evaluating Functions If students need additional help identifying functions; recognizing functions from graphs, equations, and tables; or evaluating functions; a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.

58


5

–5 –2

x


A piecewise-defined function such as  x 2 − 2, if x < −3  f (x ) = 2 x + 1, if − 3 ≤ x < 5  if x ≥ 5 8, can be graphed on a graphing calculator by following the steps below:

4. Describe the graph in Item 3 as completely as possible. Why is the graph a function?


3–4 Create Representations The graph shown in Item 3 is discontinuous at x = 0. It is important for students to note the endpoints of the intervals of the piecewise definitions. In Item 4, take special care when discussing the values of the functions at x = 0.




Answers

6. Critique the reasoning of others. Look back at Item 5. Esteban says that g(−1) = 2. Is Esteban correct? Explain. 7. Explain how to graph a piecewise-defined function. 8. If a piecewise-defined function has a break, how do you know whether to use an open circle or a closed circle for the endpoints of the function’s graph?

The domain of a piecewise-defined function consists of the union of all the domains of the individual “pieces” of the function. Likewise, the range of a piecewise-defined function consists of the union of all the ranges of the individual “pieces” of the function. You can represent the domain and range of a function by using inequalities. You can also use interval notation and set notation to represent the domain and range. 9. Write the domain and range of g(x) in Item 5 by using: a. inequalities

MATH The domain of a function is the set of input values for which the function is defined. The range of a function is the set of all possible output values for the function.

Domain: −∞ < x < ∞; range: y > 0

b. interval notation

MATH TIP

c. set notation.

Interval notation is a way of writing an interval as a pair of numbers, which represent the endpoints. For example, 2 < x ≤ 6 is written in interval notation as (2, 6]. Use a parenthesis if an endpoint is not included; use a bracket if an endpoint is included. In interval notation, infinity, ∞, and negative infinity, −∞, are not included as endpoints.

Domain: (−∞, ∞); range: (0, ∞)



Domain:{ x | x ∈ R }; range: { y | y ∈ R, y > 0}

10. Graph each function, and write its domain and range using inequalities, interval notation, and set notation. Show your work. a. b. −2 x + 2 if x < 1  x + 2 if x < 0 g (x ) =  f (x ) =  if x > 1  x − 2 2 x − 1 if x ≥ 0 y

6

6

4

4

2

2 5

–5

x

5

–5

–2

–2

–4

–4

–6

–6

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to graphing a piecewise-defined function.

My Notes

y


x

Set notation is a way of describing the numbers that are , or elements, of a set. For example, 2 < x ≤ 6 is written in set notation as {x | x ∈ R , 2 < x ≤ 6}, which is read “the set of all numbers x such that x is an element of the real numbers and 2 < x ≤ 6.”

10. a. Domain: −∞ < x < ∞, (−∞, ∞), { x | x ∈ R }; range: −∞ < y < ∞, (−∞, ∞), { y | y ∈ R } b. Domain: x ≠ 1, (−∞, 1) and (1, ∞), { x | x ∈ R , x ≠ 1}; range: y > −1, (−1, ∞), { y | y ∈ R , y > −1}


59

6. Yes. The function rule that applies when x = −1 is x + 3. Substituting −1 for x in the expression x + 3 gives −1 + 3 = 2; g(−1) = 2. 7. Graph the first function rule for the values of x given after the first “if ” statement. Then graph the second function rule for the values of x given after the second “if ” statement. Repeat this process if there are more than 2 function rules. 8. If the restriction on x for a function rule includes < or >, use an open circle for the endpoint. If the restriction on x for a function rule includes ≤ or ≥, use a closed circle for the endpoint. 9 Activating Prior Knowledge Defining the domain and range for the function will require students to look not only at the function rules and their restricted domains, but also at the graph for the range values. Formative information can be gathered as a result of this exercise. Use additional practice as needed. 10 Activating Prior Knowledge Be sure that students recognize that the function is not defined for x = 1 in Item 10b. Use these items to assess understanding and to start a class discussion of domain and range.

TEACHER to TEACHER Note that there are three different ways to express the domain and range of a function: (1) interval notation, (2) inequalities, and (3) set notation. While all three can be written to represent the same domain and range, each takes on a different look in expressing them. In Item 9, note the domain is all real numbers, and the range is all real numbers greater than zero. In the inequalities format and the set notation format, the variables x and y are used to represent the domain and range, whereas in interval notation, the ±∞ (infinity symbol) and parentheses are used. Set notation is the only format that uses braces { } and the vertical bar  to represent the words such that. The set notation also uses the symbols ∈ (element) and (set of all real numbers).


59



Answers 11. {x | x ∈ , x > 0} 12. The range includes the interval from 3, which is included, to ∞, which is not included. The interval notation for the range is [3, ∞). In set notation, the range is {y | y ∈ , y ≥ 3} 13. Sample answer: The graph has a break or hole at x = 2.

ASSESS Students’ answers to Lesson Practice problems will provide you with a formative assessment of their understanding of the lesson concepts and their ability to apply their learning. See the Activity Practice for additional problems for this lesson. You may assign the problems here or use them as a culmination for the activity. LESSON 4-1 PRACTICE

14. a. domain: −∞ < x < ∞, (−∞, ∞), {x | x ∈ }; range: y ≥ 0, [0, ∞), {y | y ≥ 0} y 4 2 2

–2

x

4

continued My Notes

Check Your Understanding WRITING MATH You can use these symbols when writing a domain or range in set notation. | such that ∈ is an element of R the real numbers Z the integers N the natural numbers

11. The domain of a function is all positive integers. How could you represent this domain using set notation? 12. Explain how to use interval and set notation to represent the range y ≥ 3. 13. What can you conclude about the graph of a piecewise-defined function whose domain is {x | x ∈ R, x ≠ 2}?

LESSON 4-1 PRACTICE 14. Graph each piecewise-defined function. Then write its domain and range using inequalities, interval notation, and set notation.  x 2 if x ≤ 0 a. f (x ) =  1  x if x > 0  2

if x < −1 3x b. f (x ) =  −x + 2 if x ≥ −1

15. The range of a function is all real numbers greater than or equal to −5 and less than or equal to 5. Write the range of the function using an inequality, interval notation, and set notation. 16. Evaluate the piecewise function for x = −2, x = 0, and x = 4. −4 x if x < −2  g (x ) = 3x + 2 if − 2 ≤ x < 4   x + 4 if x ≥ 4 17. Model with mathematics. An electric utility charges residential customers a $6 monthly fee plus $0.04 per kilowatt hour (kWh) for the first 500 kWh and $0.08/kWh for usage over 500 kWh. a. Write a piecewise function f(x) that can be used to determine a customer’s monthly bill for using x kWh of electricity. b. Graph the piecewise function. c. A customer uses 613 kWh of electricity in one month. How much should the utility charge the customer? Explain how you determined your answer.


b. domain: −∞ < x < ∞, (−∞, ∞), {x | x ∈ }; range: y ≤ 3, (−∞, 3], {y | y ≤ 3} y 4 2 –4

2

–2

4

x

–2

–6

ADAPT Check students’ answers to the Lesson Practice to ensure that they understand how to graph and evaluate piecewisedefined functions and how to identify the domain and range. If students have difficulty with the endpoints in Item 14b, encourage them to evaluate the function at x = −1 to determine which endpoint is closed and which is open.

60

® 5], Monthly Electric Bill 15. ≤ y ≤ 5, [−5, 60−5SpringBoard Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions y {y | −5 ≤ y ≤ 5} 17. b. 56 16. g(−2) = −4, g(0) = 2, g(4) = 8 48 if 0 ≤ x ≤ 500 6 + 0.04 x 17. a. f (x ) =  40 26 + 0.08(x − 500) if x > 500

Monthly Bill ($)

–4

32

24 16 8 100

300

500

700

Electricity Used (kWh)

x

c. $35.04; Sample explanation: The customer uses 613 kWh, so x = 613. This value of x satisfies the domain restriction of the second rule of the piecewise function, so I evaluated the second rule for x = 613.



Debrief students’ answers to these items to ensure that they understand different notations for the domain and range of piecewise functions.

–4


ACTIVITY 4





Lesson 4-2 Step Functions and Absolute Value Functions


PLAN

My Notes

Learning Targets:

• Graph step functions and absolute value functions. • Describe the attributes of these functions.

Pacing: 1 class period Chunking the Lesson #1–3 #4–5 #6–7 #8–9 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Interactive Word Wall, Create Representations, Look for a Pattern, Quickwrite, Think-Pair-Share

A step function is a piecewise-defined function whose value remains constant throughout each interval of its domain.

MATH

TEACH

A piecewise-defined function with a constant value throughout each interval of its domain is called a step function.

−2 if x < −3  if − 3 ≤ x < 2. 1. Graph the step function f (x ) = 1  if x ≥ 2 3 y 6 4

–4

2

–2

4

6

x

–2 –4 –6

1–3 Activating Prior Knowledge, Discussion Groups, Think-PairShare Ask students to reflect back on the word constant from earlier algebra courses and define in their own words what this means and how it applies to the step function definition. Because the pieces of a step function remain constant, each “step” is what type of line segment? When students are finished discussing, ask a representative from one group or pair to share their responses with the class and ask other students for .

2. Describe the graph in Item 1 as completely as possible.

Sample answer: The domain of the graph is the set of all real numbers,


{ x | x ∈ R }. The interval of the graph to the left of (but not including)


Bell-Ringer Activity Have students evaluate the piecewise function −x , if x < −3  f (x ) = 2 x + 1, if − 3 ≤ x < 2  if x ≥ 2  x + 3, for each value. 1. x = −4 [f(x) = 4] 2. x = −3 [f(x) = −5] 3. x = 0 [f(x) = 1] 4. x = 2 [f(x) = 5] 5. x = 5 [f(x) = 8]

2 –6

Lesson 4-2

x = −3 has a constant y-value of −2. The interval of the graph to the right of x = −3 and including x = −3, and to the left of (but not including) x = 2, has a constant y-value of 1. The interval of the graph to the right of and including x = 2 has a constant value of 3.


3. Reason abstractly. Why do you think the type of function graphed in Item 1 is called a step function?

While it is a specific type of piecewisedefined function, a step function is easy to recognize because it looks like a series of horizontal steps. Point out that a real staircase also has vertical pieces connecting the horizontal landings. This is not the case with a step function because if there were any vertical pieces, it would not be a function.

Sample answer: The graph looks like a series of steps. Each break in the graph is a “jump” up or down to the next step.


61


61



My Notes

One step function is the greatest integer function, written f(x) = _x_, which yields a value f(x) that is the greatest integer less than or equal to the value of x. For example, f(2.7) = _2.7_ = 2 because the greatest integer less than or equal to 2.7 is 2; and f(−3.1) = _−3.1_ = −4 because the greatest integer less than or equal to −3.1 is −4.

TECHNOLOGY TIP Before graphing a step function on a graphing calculator, go to the Mode window, highlight Dot, and press ENTER . Graphing a step function in dot mode will prevent the calculator from connecting breaks in the graph with line segments.

4. Graph the greatest integer function on a graphing calculator. To do so, you will need to enter the function as y = int(x). To locate int on the calculator, press MATH to reach the Math menu. Then use the right arrow key to access the Number submenu. Finally, select 5: int(. Check students’ work.

5. Make sense of problems. Describe the graph of the greatest integer function as completely as possible.

Sample answer: Each step is a horizontal segment with the left endpoint


included but the right endpoint not included. Each step is 1 unit long and 1 unit above the step to its left. The step from x = 0 (included) to x = 1 (not included) has a y-value of 0.

Differentiating Instruction Extend students’ learning by asking students to journal or write some situation in the real world that could represent a step function. After giving them some time to think and write, ask for some of their examples. Responses will vary. Below are two examples: 1. The labor cost of hiring an electrician to come to one’s home to do some work. The electrician charges a fee of $75 per hour (or any fractional part of one hour) for labor. So, the fee for 0 ≤ h ≤ 1 = $75; the fee for 1 < h ≤ 2 = $150, and so on. 2. The cost to ship a parcel has a flat rate of $5, plus an additional cost of $0.20 per ounce up to, but not exceeding, the first 12 ounces. Then, when the weight exceeds 12 ounces, the cost per ounce increases to $0.30 per ounce.

Now take a look at a different type of piecewise-defined function. −x if x < 0 f (x ) =  if x ≥ 0  x x

f(x)

−3

3

−2

2

−1

1

0

0

1

1

2

2

3

3



y 4 2 5

–5 –2 –4

x


6. Complete the table and graph the piecewise-defined function.

62

62




4–5 Chunking the Activity, Predict and Confirm, Discussion Groups Have students work with a partner. Before they use their calculators, ask students to predict what the function of y = int(x) will look like. Ask them to think about the places, or parts of the steps, that will be included (closed circle plot) and not included (an open circle plot). Then have the students follow the steps given to them to graph this function. Be sure to point out the Technology Tip in the margin for changing mode from “Connected” to “Dot,” in order to keep the calculator from connecting the breaks between the steps and making it appear that this is not really a function at all. Have them compare their predictions to the results on the calculator.





7. Reason quantitatively. Look back at the graph of f(x) shown in Item 6. a. What are the domain and range of the function?

My Notes

domain: { x | x ∈ R }; range: { y | y ∈ R, y ≥ 0}

b. Does the function have a minimum or maximum value? If so, what is it? Yes, the function has a minimum of 0.

Differentiating Instruction Students may encounter the function x f (x ) = in this or future math x classes. Use this opportunity to create the piecewise representation  f (x ) = −1, if x < 0 of f (x ) =  . if x > 0  f (x ) = 1

c. What are the x-intercept(s) and y-intercept of the function? x-intercept: 0; y-intercept: 0

d. Describe the symmetry of the graph of the function.

The graph is symmetrical about the y-axis, with an overall “V” shape.

The function f(x) in Item 6 is known as the absolute value function. The notation for the function is f(x) = |x|. The sharp change in the graph at x = 0 is the vertex.



8. Use the piecewise definition of the absolute value function to evaluate each expression. a. f(−14) = 14 b. f(8) = 8 c. f(0) = 0

(

6–7 Create Representations, Quickwrite, Think-Pair-Share In graphing this piecewise-defined function, students are introduced to the absolute value function. It is likely that students will have an informal understanding of absolute value, but it is imperative that they also understand absolute value as a piecewise-defined function for future studies in mathematics.

Then use the definition of the absolute value function to make the necessary transformation: x = −x = −1 for x < 0 and x x x = x = 1 for x > 0. Note that the x x function is undefined at x = 0.

MATH TIP The absolute value function f(x) = |x| is defined by f (x) =

{−xx

if x < 0 if x ≥ 0

)

d. f 2 − 5 = −2 + 5 CONNECT TO AP

9. Could you have determined the values of the function in Item 8 another way? Explain. Sample answer: Yes, determine the distance from zero.

The vertex of an absolute value function is an example of a cusp in a graph. A graph has a cusp at a point where there is an abrupt change in direction.


8–9 Marking the Text, Interactive Word Wall Evaluating absolute values should not be a new experience for students, but understanding this concept in of piecewise-defined functions is likely to be new. Item 9 addresses that issue.

63


63




ACTIVITY 4

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to step functions and absolute value functions.

continued My Notes


Answers

10. Construct viable arguments. Explain why the absolute value function f (x) = |x| is a piecewise-defined function.

10. The absolute value function can be defined using different rules for 2 nonoverlapping intervals of its domain. For the interval x < 0, f(x) = −x. For the interval x ≥ 0, f(x) = x. 11. The rules that define a step function are all constant functions. 12. Sample answer: Two equations must be solved, representing the two intervals of the domain.

11. How is a step function different from other types of piecewise-defined functions? 12. How does the definition of absolute value as a piecewise-defined function relate to the method of solving absolute value equations?

LESSON 4-2 PRACTICE 13. A step function known as the ceiling function, written g(x) = x , yields the value g(x) that is the least integer greater than or equal to x. a. Graph this step function. b. Find g(2.4), g(0.13), and g(−8.7).

ASSESS

Make sense of problems and persevere in solving them. A day ticket for a ski lift costs $25 for children at least 6 years old and less than 13 years old. A day ticket for students at least 13 years old and less than 19 years old costs $45. A day ticket for adults at least 19 years old costs $60. Use this information for Items 14 and 15.


14. Write the equation of a step function f(x) that can be used to determine the cost in dollars of a day ticket for the ski lift for a person who is x years old.


Use the absolute value function h(x) = |x + 2| for Items 16−19.

LESSON 4-2 PRACTICE

16. Graph the absolute value function.

13. a.

17. What are the domain and range of the function?

y


15. Graph the step function you wrote in Item 14.

18. What are the coordinates of the vertex of the function’s graph?

4

19. Write the equation for the function using piecewise notation.

3 2 1 –4

–3

–2

1

–1

2

3

4

x


–1 –2 –3 –4

b. g(2.4) = 3, g(0.13) = 1, g(−8.7) = −8

ADAPT

64

64

SpringBoard® Mathematics Algebra 2, Unit 1 • Equations, Inequalities, Functions 16.

25  14. f (x ) =  45  60

if 6 ≤ x < 13 if 13 ≤ x < 19 if x ≥ 19

y

6

4

y

15. Cost of Day Ticket ($)

Check students’ answers to the Lesson Practice to ensure that they understand concepts related to step functions and absolute-value functions. When graphing step functions, watch for students who use closed circles for all of the endpoints. Remind these students that in order for these graphs to represent functions, they must the vertical line test – no vertical line can intersect the graph in more than one point. Demonstrate that if all of the endpoints are closed, the graph will not this test.

2

60 50

–6

40 30 20 10 4

8

12 16 20 24

Age

x

–4

–2

2

17. domain: {x | x ∈ }; range: {y | y ∈ , y ≥ 0} 18. (−2, 0) if x < −2 −x − 2 19. h(x ) =  if x ≥ −2  x + 2


4

6

x




Lesson 4-3 Transforming the Absolute Value Parent Function


PLAN

My Notes

Learning Targets:

the effect on the graph of replacing f(x) by f(x) + k, k ⋅ f(x), • Identify f(kx), and f(x + k). • Find the value of k, given these graphs.

Pacing: 1 class period Chunking the Lesson #1 #2–3 #4–5 #6 #7–8 #9–10 #11 Example A Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Create Representation, Look for a Pattern, Debriefing, Think-Pair-Share, Identify a Subtask

The absolute value function f(x) = |x| is the parent absolute value function. Recall that a parent function is the most basic function of a particular type. Transformations may be performed on a parent function to produce a new function.

TEACH Bell-Ringer Activity Students should be familiar with how to evaluate absolute value functions. Have students evaluate each function for the given value: 1. f(x) = |x| for x = 9 [f(x) = 9] 2. f(x) = |x| for x = −9 [f(x) = 9] 3. f(x) = |x + 1| for x = 9 [f(x) = 10] 4. f(x) = |x + 1| for x = −9 [f(x) = 8]

1. Model with mathematics. For each function below, graph the function and identify the transformation of f(x) = |x|. a. g(x) = |x| + 1 b. h(x) = |x| − 2 y

f(x) = |x|

g(x) = |x| + 1

y

f(x) = |x|

4

4

2

2 5

–5

x

5

–2

–2

–4

–4

vertical translation up 1 unit


h(x) = |x| – 2

d. q(x) = −| x | y

y k(x) = 3|x|


x

vertical translation down 2 units

c. k(x) = 3|x|

f(x) = |x|

MATH TIP

–5

4

4

2

2 5

–5

x

–5

–2

–2

–4

–4

vertical stretch by a factor of 3

f(x) = |x|

5 q(x) = –|x|

Lesson 4-3

x

Transformations include: • vertical translations, which shift a graph up or down • horizontal translations, which shift a graph left or right • reflections, which produce a mirror image of a graph over a line • vertical stretches or vertical shrinks, which stretch a graph away from the x-axis or shrink a graph toward the x-axis • horizontal stretches or horizontal shrinks, which stretch a graph away from the y-axis or shrink a graph toward the y-axis

Students should discuss the impact of the “+ 1” in Items 3 and 4. 1 Activating Prior Knowledge, Create Representations With text-based reminders about parent functions and transformations, students are asked to identify transformations of the parent absolute value function. Transformations that students are expected to recognize from earlier math courses include the following: vertical translations, vertical stretches/shrinks, and reflections over the x-axis.


reflection over the x-axis


MINI-LESSON: Vertical Translations and Vertical Stretch/Shrink If students need additional help graphing vertical translations of parent functions or vertical stretches or shrinks of a parent function, a mini-lesson is available to provide practice.

65

A parent function is the most basic function of a particular type. This lesson refers to the parent absolute value function as f(x) = |x|. Here are a few more examples of common parent functions: Linear: f(x) = x, or y = x Quadratic: f(x) = x2, or y = x2 Cubic: f(x) = x3, or y = x3 Inverse: f (x )= 1x , or y = 1 x Radical: f (x )= x , or y = x Each of these parent functions can be altered to change its basic position, size, and shape. When this occurs, a new function, known as a transformation, is produced. The cause of a transformation involves some type of arithmetic operation(s) to the parent function.

See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.


65




ACTIVITY 4

My Notes

TECHNOLOGY TIP Many graphing calculators use a function called “abs” to represent absolute value.

2. Use the coordinate grid at the right. a. Graph the parent function f(x) = |x|.

y

h(x) = |–2x|

b. Predict the transformation for g(x) = |x − 3| and h(x) = |−2x|.

4

g(x) = |x – 3| 5

–5

c. Graph the function g(x) = |x − 3| and h(x) = |−2x|.

2–3 Create Representations, Predict and Confirm Horizontal translations are introduced in Item 2 and then expanded in Items 3–5.

x

–2 –4

d. What transformations do your graphs show?

g(x) is a horizontal translation of f(x) 3 units to the right. h(x) is a

4–5 Look for a Pattern The counterintuitive nature of this translation, x − a moving to the right and x + a moving to the left, may be a surprise to students. Items 4 and 5 provide the opportunity to reason abstractly to generalize the effects of x ± a.

vertical stretch of f(x) by a factor of 2.

3. Reason abstractly and quantitatively. Use the results from Item 2 to predict the transformation of h(x) = |x + 2|. Then graph the function to confirm or revise your prediction. Sample prediction: horizontal translation 2 units to the left y 4

TEACHER to TEACHER

2 h(x) = |x + 2|

Here is one way of explaining horizontal translations given by x − a: Show students that the y-value for the graph of y = f(x − a) at a is the same y-value that y = f(x) has at 0. When x = a, then y = f(x − a) = f(a − a) = f(0). When x = 0, then y = f(x) = f(0). The y-values are equal.

–5

f(x) = |x|

5

x

–2 –4

The functions in Items 2 and 3 are examples of horizontal translations. A horizontal translation occurs when the independent variable, x, is replaced with x + k or with x − k.

Point out that those functions show a horizontal shift in the x-values from 0 to a. If a is positive, this shift is to the right a units, and if a is negative, this shift is to the left a units. In general, the y-value for the graph of y = f(x − a) at b + a is the same y-value that y = f(x) has at x = b. Again, point out that the shift is a units right if a > 0 and left if a < 0.

4. In the absolute value function f(x) = |x + k| with k > 0, describe how the graph of the function changes, compared to the parent function. The function moves k units to the left.

5. In the absolute value function f(x) = |x − k| with k > 0, describe how the graph of the function changes, compared to the parent function. The graph moves k units to the right.

66


MINI-LESSON: Reflections over the x-axis If students need additional help graphing reflections of parent functions over the x-axis, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.

66

2

f(x) = |x|

Predictions may vary.



If students experience difficulties in describing transformations, or if they have limited experience with transforming parent functions, then giving them additional opportunities to graph transformations may be appropriate. The practice exercises on this page and the next may be helpful in reinforcing understanding.

continued









b. f(x) = |x + 5|

y 4

2

–2

1. Press the y = key in the upper left corner. 2. Press the CLEAR key to clear any previous functions. 3. Press the MATH key. 4. Press the right arrow one time to highlight the NUM command. 5. Select option 1, “abs(” for absolute value, by pressing ENTER . 6. Press the X,T,Θ,n key.

y

4

–5

To graph the parent absolute value function on a TI graphing calculator, follow these basic steps:

My Notes

6. Graph each function. a. f(x) = |x − 4|

2

x

5 f(x) = |x – 4|

x

–5 f(x) = |x + 5|

–2

–4

–4

y

7. Use the coordinate grid at the right. a. Graph the parent function f(x) = |x| and the function g(x) = |2x|. b. Describe the graph of g(x) as a horizontal stretch or horizontal shrink of the graph of the parent function.

6

MATH TIP f(x) = |x| 5

–5

x

–2

a horizontal shrink by a 1 factor of 2

7. Press the ) key to close parentheses. 8. Press the GRAPH key.

g(x) = |2x|

4 2

8. Express regularity in repeated reasoning. Use the results from

A horizontal stretch or shrink by a factor of k maps a point (x, y) on the graph of the original function to the point (kx, y) on the graph of the transformed function. Similarly, a vertical stretch or shrink by a factor of k maps a point (x, y) on the graph of the original function to the point (x, ky) on the graph of the transformed function.

Item 7 to predict how the graph of h(x) = 1 x is transformed from the



2

graph of the parent function. Then graph h(x) to confirm or revise your prediction. Sample prediction: a horizontal stretch by a factor of 2 y 6

f(x) = |x|

4 h(x) = 2 5

–5

Technology Tip

1 x 2

Note: Use this basic process with the transformations of the parent absolute value function. For additional technology resources, visit SpringBoard Digital. 6 Predict and Confirm, Create Representations, Chunking the Activity Before students graph the functions given in Item 6, have them write down their predictions for the transformation that will take place with the parent absolute value graph. Have them confirm their predictions by graphing. Before moving on to the next items, sketch a couple of absolute value horizontal transformations, and ask students to write the equations of the functions you graphed. 7–8 Discussion Groups, Predict and Confirm, Critique Reasoning, Think-Pair-Share After presenting Items 7 and 8, have students work in small groups or with a partner to try to predict how the value of k in f (x ) = kx affects the graph of f(x) = |x|. In other words, what if g (x ) = 3x and h(x ) = 1 x ? Have students collaborate 3 and share their findings.

x

–2


67


67



ACTIVITY 4 Continued 9–10 Close Reading, Marking the Text Students probably will not have too much difficulty expressing regularity in repeated reasoning and coming up with a generalization for Item 9, based upon the previous items. However, Item 10 is written in such a way that you may want to emphasize some points. The functions in Items 9 and 10 are written the same, but students should look closely at the values of k. In Item 10, k is really a fractional value. Because of the way this is written, the only difference between the answers for Items 9 and 10 is the words shrink and stretch. The reason why the factors are both 1 is because 1 , k k when k is a fraction, is the inverse of the fractional value.



9. In the absolute value function f(x) = |kx| with k > 1, describe how the graph of the function changes compared to the graph of the parent function. What if k < −1? The graph of f(x) is a horizontal shrink of the graph of the parent 1 function by a factor of k when k > 1 or when k < −1.

10. In the absolute value function f(x) = |kx| with 0 < k < 1, describe how the graph of the function changes compared to the graph of the parent function. What if −1 < k < 0? The graph of f(x) is a horizontal stretch of the graph of the parent 1 function by a factor of k when 0 < k < 1 or when −1 < k < 0.

11. Each graph shows a transformation g(x) of the parent function f(x) = |x|. Describe the transformation and write the equation of g(x). y a. g(x)

11 Summarizing Point out that the phrases “vertical stretch” and “horizontal shrink” can both be used to describe what is taking place in Item 11a. However, the difference between these descriptions is the values of their factors.

f(x) = |x|

6 4 2 5

–5

x

–2

f(x) = |x|

4 g(x) 2 5

–5

x

–2 –4

vertical translation down 3 units; g(x) = |x| − 3

68

68




y

b.


1 vertical stretch by a factor of 3 or horizontal shrink by a factor of 3 ; g(x) = 3|x| or g(x) = |3x|






Example A

Describe the transformations of g(x) = 2|x + 3| from the parent absolute value function and use them to graph g(x). Step 1: Describe the transformations. g(x) is a horizontal translation of f(x) = |x| by 3 units to the left, followed by a vertical stretch by a factor of 2.

Step 2:

Apply the horizontal translation first, and then apply the vertical stretch. Apply the horizontal translation. Graph f(x) = |x|. Then shift each point on the graph of f(x) by 3 units to the left. To do so, subtract 3 from the x-coordinates and keep the y-coordinates the same. Name the new function h(x). Its equation is h(x) = |x + 3|. y

Example A Activating Prior Knowledge This concept of unraveling multiple transformations to a parent absolute value function is similar to following orders of operations with arithmetic operations.

MATH TIP

Try These A

To graph an absolute value function of the form g(x) = a|b(x − c)| + d, apply the transformations of f(x) = |x| in this order: 1. horizontal translation 2. reflection in the y-axis and/or horizontal shrink or stretch 3. reflection in the x-axis and/or vertical shrink or stretch 4. vertical translation

Answers a.

y 4 2

–2

6

–4 f(x) = |x|

b.

2 –6

–4

2

–2

y

x

4

–2



Step 3:

x

5

–5

h(x) = |x + 3|

4

–8

h(x) = –|x – 1| + 2

4 2

Apply the vertical stretch.

Now stretch each point on the graph of h(x) vertically by a factor of 2. To do so, keep the x-coordinates the same and multiply the y-coordinates by 2. Solution: The new function is g(x) = 2|x + 3|. y

g(x) = 2|x + 3|

TECHNOLOGY TIP You can check that you have graphed g(x) correctly by graphing it on a graphing calculator.

5

–5 –2

k(x) = 4|x + 1| – 3 –4

h(x) = |x + 3|

6 4

Universal Access

2 –8

–6

–4

2

–2

4

For students having difficulty with the concept of absolute value, explain that an absolute value represents a distance, or measure, of a number from the origin of a number line. Furthermore, a measurement cannot be a negative value.

x

–2

Try These A For each absolute value function, describe the transformations represented in the rule and use them to graph the function. a. h(x) = −|x − 1| + 2 translate to the right 1 unit, reflect over the x-axis, and translate up 2 units

b. k(x) = 4|x + 1| − 3 translate 1 unit to the left, vertically stretch by a factor of 4, then translate 3 units down


69

Keeping that in mind, when applying this concept to absolute value functions and their graphs, this is why they remain above the x-axis unless written with a negative sign preceding the absolute value bars.


69

x


ACTIVITY 4 Continued Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to transformations of functions.




Answers 12. g(x) and f(x) are the same function. This relationship makes sense because |x| = |−x|.

12. Graph the function g(x) = |−x|. What is the relationship between g(x) and f(x) = |x|? Why does this relationship make sense? 13. Compare and contrast a vertical stretch by a factor of 4 with a horizontal stretch by a factor of 4.

y

6

14. Without graphing the function, determine the coordinates of the vertex of f(x) = |x + 2| − 5. Explain how you determined your answer.

4 2

2

4

6

x

LESSON 4-3 PRACTICE

13. Sample answer: Both transformations stretch points on the original graph away from an axis. A vertical stretch maps a point (x, y) on the original graph to point (x, 4y) on the transformed graph. A horizontal stretch maps a point (x, y) on the original graph to point (4x, y) on the transformed graph. 14. (−2, −5); Sample explanation: The graph of f(x) is a translation of the graph of the absolute value parent function by 2 units left and 5 units down. The vertex of the graph of the absolute value parent function is (0, 0), so the vertex of the graph of f(x) must be (−2, −5).

15. The graph of g(x) is the graph of f(x) = |x| translated 6 units to the right. Write the equation of g(x). 16. Describe the graph of h(x) = −5|x| as one or more transformations of the graph of f(x) = |x|. 17. What are the domain and range of f(x) = |x + 4| − 1? Explain. 18. Graph each transformation of f(x) = |x|. a. g(x) = |x − 4| + 2 b. g(x) = |2x| − 3 c. g(x) = −|x + 4| + 3 d. g(x) = −3|x + 2| + 4 19. Attend to precision. Write the equation for each transformation of f(x) = |x| described below. a. Translate left 9 units, stretch vertically by a factor of 5, and translate down 23 units. b. Translate left 12 units, stretch horizontally by a factor of 4, and reflect over the x-axis. y c. 2

ASSESS

5


–6 –4 –2

x

–2


–4



15. g(x) = |x − 6| 16. a reflection over the x-axis and a vertical stretch by a factor of 5 17. Domain: {x | x ∈ }; range: {y | y ∈ , y ≥ −1}; Sample explanation: The function is defined for all real values of x, so the domain is all real numbers. The graph of f(x) opens upward, and its vertex is at (−4, −1). −1 is the minimum value of f(x), so the range is all real numbers ≥ −1.

ADAPT Check students’ answers to the Lesson Practice to ensure that they understand transformations of the absolute value parent function. If students have trouble graphing a function by using the values in the equation, allow them to graph by generating ordered pairs.

70

d. y 18. y 70a. SpringBoard® Mathematics Algebra 2, Unit 1c.• Equations, Inequalities, Functions

y

4

4

4

2

2

2

5

–5

x

y

b. 4 2

5

–5 –2

x

x

–5

x

–5

–2

–2

–4

–4

19. a. f(x) = 5|x + 9| − 23 1 b. f (x ) = − 4 (x + 12) c. f(x) = 3|x − 2| − 4 or f(x) = |3(x − 2)| − 4





Piecewise-Defined Functions Absolutely Piece-ful


ACTIVITY 4 PRACTICE

8. Evaluate f(x) for x = −4, x = 1, and x = 4.


b.

y

if x < −3 −5x  f (x ) =  x 2 if − 3 ≤ x < 4  2 x + 4 if x ≥ 4

Lesson 4-1 1. Graph each of the following piecewise-defined functions. Then write its domain and range using inequalities, interval notation, and set notation. a. −3x − 4 if x < −1 f ( x ) =   x if x ≥ −1

ACTIVITY PRACTICE 1. a. domain: −∞ < x < ∞, (−∞, ∞), {x | x ∈ }; range: y ≥ −1, [−1, ∞), {y | y ≥ −1} 6 4 2

9. Write the equation of the piecewise function f(x) shown below.

–6

y 4

 x 2 if x ≤ 1 f (x ) =  −2 x + 3 if x > 1

2

–2

x

4

–2

2. Explain why the graph shown below does not represent a function.

Lesson 4-2

2

10. a. Graph the step 2

4

–4

–4

3. Write a piecewise function f(x) that can be used to determine the welder’s earnings when she works x hours in a week. 4. Graph the piecewise function. 5. How much does the welder earn when she works 48 hours in a week? A. $990 B. $1040 C. $1200 D. $1440

2

–2

x

4

–2 –4

b. What are the domain and range of the step function?

A welder earns $20 per hour for the first 40 hours she works in a week and $30 per hour for each hour over 40 hours. Use this information for Items 3–5.

11. It costs $30 per day or $90 per week to rent a wallpaper steamer. If the time in days is not a whole number, it is rounded up to the nextgreatest day. Customers are given the weekly rate if it is cheaper than using the daily rate. a. Write the equation of a step function f(x) that can be used to determine the cost in dollars of renting a wallpaper steamer for x days. Use a domain of 0 < x ≤ 7. b. Graph the step function.

2. When x = −2, y = 2, and y = −2; because the input −2 has 2 outputs, the relationship is not a function. 3. f (x ) =

20 x {800 + 30( x − 40)

4. y

if 0 ≤ x ≤ 40 if x > 40

Weekly Earnings

800

Earnings ($)

6. The domain of a function is all real numbers greater than −2 and less than or equal to 8. Write the domain using an inequality, interval notation, and set notation. 7. The range of a function is [4, ∞). Write the range using an inequality and set notation.

640 480 320 160 8

16

24

32

40

x

Hours Worked

10. a.

30 if 0 < x ≤ 1 Functions Activity  4 • Piecewise-Defined if 1 < x ≤ 2 11. a. f (x ) = 60  if 2 < x ≤ 7 90

y 4

–4

y

2

–2

4

–4

5. B 6. −2 < x ≤ 8, (−2, 8], {x | x ∈ , −2 < x ≤ 8} 7. y ≥ 4; {y | y ∈ , y ≥ 4} 8. f(−4) = 20, f(1) = 1, f(4) = 12 −2 x 9. f (x ) =   x − 2

if x < 1 if x ≥ 1

100

x

–2

71

Cost of Renting Wallpaper Steamer

b.

2

80

Cost ($)


2

 4 if x < −2  function f (x ) = 1 if − 2 ≤ x < 3 .  −3 if x ≥ 3

x

–2


x

6

4

4

–2

4

y

–4

y

–4

2

–2

b. domain: −∞ < x < ∞, (−∞, ∞), {x | x ∈ }; range: −∞ < y < ∞, (−∞, ∞), {y | y ∈ }

2 –4

–4

60 40 20

b. domain: {x | x ∈ }; range: {y | y = 4, y = 1, or y = −3}

1

2

3

4

5

6

7

x

Days Rented


71



ACTIVITY 4 Continued 12. a.

Piecewise-Defined Functions Absolutely Piece-ful

ACTIVITY 4

y

continued

4

4

–2

13. A step function called the nearest integer function gives the value g(x) that is the integer closest to x. For half integers, such as 1.5, 2.5, and 3.5, the nearest integer function gives the value of g(x) that is the even integer closest to x. a. Graph the nearest integer function. b. Find g(−2.1), g(0.5), and g(3.6).

–4

b. −2; 0; 3 13. a.

y 4 2 –4

2

–2

x

4

–2

b. −2; 0; 4 −1 if x < 0 14. f (x ) =  if x > 0 1 y

1 2

x

16. Write the equation of the function g(x) shown in the graph, and describe the graph as a transformation of the graph of f(x) = |x|. y

–2

15. a. Check students’ graphs. b. domain: {x | x ∈ }; range: {y | y ∈ , y ≥ −1} c. x-intercepts: −3 and −1; y-intercept: 1 d. The graph is symmetric about the vertical line x = −2. 16. g(x) = −2|x| + 3 or g(x) = −|2x| + 3; The graph of g(x) is a reflection of the graph of f(x) across the x-axis, followed by a vertical stretch by a factor of 2, and then a translation 3 units up. 17. a. a horizontal translation 3 units left and a vertical translation 1 unit down b. a vertical shrink by a factor of 1 followed by a translation 3 2 units up c. a translation 1 unit right, followed by a reflection across the x-axis and a vertical stretch by a factor of 2, and then a translation 1 unit down d. a translation 1 unit right, followed by a horizontal shrink by a factor of 1 , and then a 5 translation 4 units down ADDITIONAL PRACTICE If students need more practice on the concepts in this activity, see the Teacher Resources at SpringBoard Digital for additional practice problems.

72

19. Which function is shown in the graph? y 6

14. Use the definition of f(x) = |x| to rewrite |x| f (x ) = as a piecewise-defined function. x Then graph the function.

Lesson 4-3

2

1

18. Write the equation for each transformation of f(x) = |x| described below. a. translated right 7 units, shrunk vertically by a factor of 0.5, and translated up 5 units b. stretched horizontally by a factor of 5, reflected over the x-axis, and translated down 10 units c. translated right 9 units and translated down 6 units

4 2

15. Consider the absolute value function f(x) = |x + 2| − 1. a. Graph the function. b. What are the domain and range of the function? c. What are the x-intercept(s) and y-intercept of the function? d. Describe the symmetry of the graph.

–4

–2 –1 –1

12. A step function called the integer part function gives the value f(x) that is the integer part of x. a. Graph the integer part function. b. Find f(−2.1), f(0.5), and f(3.6).

4 2 –4

2

–2

4

x

–2 –4

17. Graph the following transformations of f(x) = |x|. Then identify the transformations. a. g(x) = |x + 3| − 1 b. g(x) = 1 |x| + 2 3 c. g(x) = −2|x − 1| − 1

–4

2

–2

4

x

–2

A. f(x) = |x − 2| + 1 B. f(x) = |x − 1| + 2 C. f(x) = |x + 2| + 1 D. f(x) = |x + 1| + 2


Reason Abstractly and Quantitatively 20. Before answering each part, review them carefully to ensure you understand all the terminology and what is being asked. a. Describe how the graph of g(x) = |x| + k changes compared to the graph of f(x) = |x| when k > 0 and when k < 0. b. Describe how the graph of h(x) = k|x| changes compared to the graph of f(x) = |x| when k > 1 and when 0 < k < 1. c. Describe how the graph of j(x) = |kx| changes compared to the graph of f(x) = |x| when k < 0.

d. g(x) = 5|x − 1| − 4

y Algebra 2, Unit 1 • Equations, 17. 72a–d. SpringBoard® Mathematics Functions 18.Inequalities, a. f(x) = 0.5|x − 7| + 5 1 x − 10 f ( x ) = − b. d 6 5 c. f(x) = |x − 9| − 6 b 19. A 4 20. a. When k > 0, the graph of g(x) is the graph of f(x) translated k units up. When k < 0, the 2 a graph of g(x) is the graph of f(x) translated x k units down. 2 4 6 –6 –4 –2 b. When k > 1, the graph of h(x) is the graph of –2 f(x) vertically stretched by a factor of k. When 0 < k < 1, the graph of h(x) is the graph of f(x) –4 vertically shrunk by a factor of k. c. When k < 0, the graph of j(x) is f(x) horizontally –6 c stretched or shrunk by a factor of 1 . k



2

–4

x


2



ACTIVITY

Function Composition and Operations

ACTIVITY 5

New from Old Lesson 5-1 Operations with Functions

5

Investigative Activity Standards Focus In Activity 5, students perform operations on functions. Students then write composite functions. Throughout this activity, emphasize that when evaluating functions combined with operations, the value of an input evaluated first in the separate functions and then operated is equal to the value of the combined function with that input. Combining functions ahead of time is efficient when evaluating many input values.

My Notes

Learning Targets:

• Combine functions using arithmetic operations. • Build functions that model real-world scenarios.

SUGGESTED LEARNING STRATEGIES: Activating Prior Knowledge, Discussion Groups, Debriefing, Close Reading, Think-Pair-Share, Summarizing, Paraphrasing, Quickwrite

Jim Green has a lawn service called Green’s Grass Guaranteed. Tori and Stephan are two of his employees. Tori earns $10 per hour, and Stephan earns $8 per hour. Jim sends Tori and Stephan on a job that takes them 4 hours. 1. Model with mathematics. Write a function t(h) to represent Tori’s earnings in dollars for working h hours and a function s(h) to represent Stephan’s earnings in dollars for working h hours.

Lesson 5-1 PLAN

t(h) = 10h; s(h) = 8h

Pacing: 1 class period Chunking the Lesson 2. Find t(4) and s(4) and tell what these values represent in this situation. t(4) = 40; s(4) = 32; Tori’s earnings for the 4-hour job are $40, and Stephan’s earnings for the 4-hour job are $32.

3. Find t(4) + s(4) and tell what it represents in this situation.

MATH TIP Addition, subtraction, multiplication, and division are operations on real numbers. You can also perform these operations with functions.

t(4) + s(4) = 40 + 32 = 72; The sum of Tori and Stephan’s earnings for the 4-hour job is $72.

TEACH Bell-Ringer Activity


You can add two functions by adding their function rules.


#1–3 #4–7 #8–11 #12 #13–14 #15 #16–17 #18–20 #21 #22 Check Your Understanding Lesson Practice

4. a. Add the functions t (h) and s (h) to find (t + s)(h). Then simplify the function rule. (t + s)(h) = 10h + 8h = 18h

b. What does the function (t + s)(h) represent in this situation?

WRITING MATH The notation (f + g)(x) represents the sum of the functions f (x) and g(x). In other words, (f + g)(x) = f (x) + g(x).

the amount in dollars Jim must spend on Tori and Stephan’s earnings for a job that takes h hours

Have students review some of the concepts they will need to apply in this lesson. Ask students to complete the following exercises. 1. Simplify 5a2 + 2a − 4a − 6a2. [−a2 − 2a] 2. Evaluate (m + 2)2 for m = −5. [9] Ask students to share their responses, and answer any questions they may have prior to moving forward with the lesson.

5. Find (t + s)(4). How does the answer compare to t(4) + s(4)?

(t + s)(4) = 18(4) = 72; (t + s)(4) has the same value as t(4) + s(4).

6. How much will Jim spend on Tori and Stephan’s earnings for the 4-hour job? $72

Activity 5 • Function Composition and Operations

73

Common Core State Standards for Activity 5 HSF-BF.A.1

Write a function that describes a relationship between two quantities.

HSF-BF.A.1b

Combine standard function types using arithmetic operations.

HSF-BF.A.1c(+) Compose functions. [Note: HSF-BF.A.1c is introduced in this activity but is also addressed in higher level mathematics courses.]

1–3 Activating Prior Knowledge, Chunking the Activity, Paraphrasing Ask students questions like: What would the graph of these functions look like? [t(h) would be linear, with a y-intercept at the origin and a slope of 10 , and s(h) 1 would be linear, with a y-intercept at the origin and a slope of 8.] What does the 1 y-intercept represent? [zero pay for zero hours worked] What does the slope represent? [the rate of pay per hour] Would you be interested in looking at the entire coordinate plane? [No; Quadrant I only, because Tori and Stephan will not receive pay for negative hours] 4–7 Discussion Groups, Group Presentation Ensure students understand this application by placing them in small groups and having each group create a scenario with adding two real-world functions. Encourage them to use the hourly earnings functions as a template but also to feel free to use a variable other than hours.


73




Lesson 5-1 Operations with Functions

ACTIVITY 5 You can use the table feature of a graphing calculator to find specific function values. You could do this for the function addition represented in the previous items by following these steps:


$108; I evaluated (t + s)(h) for h = 6: (t + s)(6) = 18(6) = 108.

For a basic tree-trimming job, Jim charges customers a fixed $25 fee plus $150 per tree. One of Jim’s competitors, Vista Lawn & Garden, charges customers a fixed fee of $75 plus $175 per tree for the same service. 8. Write a function j(t) to represent the total charge in dollars for trimming t trees by Jim’s company and a function v(t) to represent the total charge in dollars for trimming t trees by Vista. j(t) = 25 + 150t; v(t) = 75 + 175t

MATH TIP

74

9. a. Subtract j(t) from v(t) to find (v − j )(t). Then simplify the function rule. (v − j)(t) = 75 + 175t − (25 + 150t) = 75 + 175t − 25 − 150t = 50 + 25t

When subtracting an algebraic expression, to subtract each term of the expression. For example, subtract 6x − 2 from 10x as follows. 10x − (6x + 2) = 10x − 6x − 2 = 4x − 2

WRITING MATH The notation (f − g)(x) represents the difference of the functions f(x) and g(x). In other words, (f − g)(x) = f(x) − g(x).

8–11 Activating Prior Knowledge, Chunking the Activity, Predict and Confirm Lead a discussion about these items by asking the following: • How do these functions differ from those presented in Items 1−7? [These functions have a y-intercept other than (0, 0).] • What makes these functions have these y-intercepts? [the fixed fees charged by each company] • Why is subtraction being used rather than addition? [It is basically an example of comparison shopping, where one wants to know how much will be saved by using one company instead of the other, in of a given number of trees.] • What is one thing you have to be cautious about when subtracting expressions? [Subtract each term of the expression, not just the first term; in other words, the subtraction sign is distributed throughout the subtrahend expression.]

7. How much would Jim spend on Tori and Stephan’s earnings for a job that takes 6 hours? Explain how you determined your answer.

b. What does the function (v − j)(t) represent in this situation?

the amount in dollars a customer will save by choosing Jim’s company to trim t trees rather than Vista

10. Find (v − j )(5). What does this value represent in this situation? (v − j )(5) = 50 + 25(5) = 175; A customer will save $175 by choosing Jim’s company rather than Vista to trim 5 trees.

11. How much will a customer save by choosing Jim’s company to trim 8 trees rather than choosing Vista? Explain how you determined your answer. $250; I evaluated (v − j )(t) for t = 8: (v − j )(8) = 50 + 25(8) = 250.

12. Look for and make use of structure. Given f (x) = 3x + 2, g (x) = 2x − 1, and h(x) = x2 − 2x + 8, find each function and simplify the function rule. a. ( f + g)(x) = 5x + 1

b. (g + h)(x) = x2 + 7

c. (h + f )(x) = x + x + 10

d. ( f − g)(x) = x + 3

e. (g − f )(x) = −x − 3

f. (h − g)(x) = x2 − 4x + 9

2

74


12 Activating Prior Knowledge, Debriefing Once students know how to add and subtract linear functions, they apply this knowledge to adding and/or subtracting a linear function to a quadratic function. The same function rules apply, and students will use the same structure to combine like .



1. Press the y = key. 2. Beside the function, type in 10x + 8x. 3. Press 2nd WINDOW to look at the table setup. Set TblStart to =0; Set the change in the table, or ∆Tbl, to 1; Set the Indpnt: to Ask. 4. Access the table by pressing 2nd GRAPH . 5. Notice the table is blank. The calculator is waiting for you to enter the x-value (in this case, the number of hours) for which you would like to know the corresponding y-value, or cost. 6. At x=, key in 4 ENTER . This should give the corresponding y-value of $72. 7. At x=, key in 6 ENTER . This should give the corresponding y-value of $108. 8. Now you can continue this by trying other numbers of hours that were not already in the examples.

My Notes


Technology Tip

continued






Jim has been asked to make a bid for installing the shrubs around a new office building. In the bid, he needs to include the number of shrubs he can install in an 8-hour day, the cost per shrub including installation, and the total cost of his services for an 8-hour day. 13. a. Write a function n(h) to represent the number of shrubs Jim can install in an 8-hour day when it takes him h hours to install one shrub. n( h) = 8 h

b. What are the restrictions on the domain of n(h)? Explain.

The value of h cannot be 0, or the function would be undefined. Also, because h represents a number of hours, its value cannot be negative.

For those students who need additional explanation of the functions used in Item 12, explain the following:

My Notes

MATH TIP When considering restrictions on the domain of a real-world function, consider both values of the domain for which the function would be undefined and values of the domain that would not make sense in the situation.

14. Jim will charge $16 for each shrub. He will also charge $65 per hour for installation services. Write a function c(h) to represent the amount Jim will charge for a shrub that takes h hours to install.

The total cost of Jim’s services for an 8-hour day is equal to the number of shrubs he can install times the charge for each shrub.



⋅

⋅8

( n c )( h) = (16 + 65 h) = 128 + 520 h h

WRITING MATH

⋅

⋅

⋅

⋅

The value of h must be positive.

15 Activating Prior Knowledge Explain to the students that multiplying functions will require them to multiply polynomials, which they learned in Algebra 1. In Item 15a, a monomial is being multiplied by a binomial. The types of polynomials being multiplied will obviously vary with the functions. Be sure to use the Distributive Property. In Item 15b, students again encounter the topic of domain restrictions. To ensure that students understand the correct response to 15b, engage them by asking them why the h-value must be positive.

• The standard form of a linear function is Ax + By = C, where A, B, and C are constants. • The y-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. • The point-slope form of a linear equation is y − y1 = m(x − x1), where m is the slope and (x1, y1) are the coordinates of a point through which the line es.

• The general form is ax2 + bx + c = 0, where a, b, and c are constants.

The notation (f g)(x) represents the product of the functions f (x) and g(x). In other words, (f g)(x) = f (x) g(x).

b. Attend to precision. What are the restrictions on the domain of (n c)(h)?

A linear function is an algebraic equation in which the greatest degree of a variable term is 1. In other words, the greatest exponent of a variable term is 1.

A quadratic function is an algebraic equation in which one or more of the variable is squared, giving the function a degree of 2. However, a squared power is the greatest degree a quadratic function can have.

c(h) = 16 + 65h

15. a. Find the total cost of Jim’s services using the functions n(h) and c(h) to find (n c)(h). Then simplify the function rule.



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13–14 Close Reading, Marking the Text, Differentiating Instruction, Simplify the Problem To help students understand the function in Item 13, have them construct a table of values for h and n(h). Ask: If it takes Jim one hour to install one shrub, how many shrubs can Jim install in an 8-hour day? What if it takes Jim 2 hours to install one shrub? 3 hours? 4 hours? Elicit from students the operation of division between 8, the total number of hours in the workday, and h, the number of hours it takes Jim to install one shrub. In Item 13b, highlight restrictions and domain. students whose first language is not English by further explaining the word restriction. For Item 14, tell students who are struggling to refer back to either function from Item 8 because they are the same type.


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⋅

My Notes CONNECT TO BUSINESS When a company makes a bid on a job, the company states the price at which it is willing to do the job. The company must make its bid high enough to cover all of its expenses. If it bids too high, however, the job may be offered to one of its competitors.

16. Reason quantitatively. Jim estimates that it will take 0.5 hour to install each shrub. Use the functions n(h), c(h), and (n c)(h) to determine the following values for Jim’s bid, and explain how you determined your answers. a. the number of shrubs Jim can install in an 8-hour day

⋅

16 shrubs; I evaluated n(h) for h = 0.5: n(0.5) = 8 = 16. 0 .5

b. the cost per shrub, including installation

$48.50; I evaluated c(h) for h = 0.5: c(0.5) = 16 + 65(0.5) = 48.50.

c. the total cost of Jim’s services for an 8-hour day $776; I evaluated (n

+ 520 = 776. ⋅ c)(h) for h = 0.5: (n⋅c)(0.5) = 128 0 .5

17. Explain how you could check your answer to Item 16c.

Sample answer: Multiply the number of shrubs Jim can install in an 8-hour day by the cost per shrub, including installation: 16($48.50) = $776.

Differentiating Instruction Ask students to discuss whether their conjectures were correct or incorrect when they altered the value of h in Items 16 and 17 from h = 0.5 to h = 2 (a longer amount of time per 3 shrub). Why is the total cost of Jim’s services for an 8-hour day less? [because he is getting less work done per hour]

Jim offers two lawn improvement services, as described in the table. Lawn Improvement Services Service

Hourly Charge ($)

Material Cost for Average Yard ($)

Compost

40

140

Fertilizer

30

30

18. a. Write a function c(h) to represent the total charge for applying compost to a lawn, where h is the number of hours the job takes. c(h) = 40h + 140

b. Write a function f (h) to represent the total charge for applying fertilizer to a lawn, where h is the number of hours the job takes. f (h) = 30h + 30

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Present a table with four column titles—h, n(h), c(h), and (n c)(h)—and place values for h = 0.5 in each column. Then have students make a conjecture as to whether they think it will cost more or less if Jim estimates that it will take him 40 minutes to install each shrub. After discussing, have students try h = 2 in the functions. Fill in values 3 for h = 2 in a new row of the table. 3 [Answer should be approximately $712.]




16–17 Create Representations, Debriefing In Item 16, ensure students are not confused by the solution to 16a being 16 shrubs and the predetermined cost per shrub (listed in Item 14) being the same. The answer of 16 in Item 16b represents that predetermined cost per shrub. The fact that the number of shrubs and the cost per shrub are the same is a mere coincidence.






19. a. Divide c(h) by f (h) to find (c ÷ f )(h) given that f (h) ≠ 0. ( c ÷ f )( h) = 40 h + 140 30 h + 30

My Notes

WRITING MATH

b. What does the function (c ÷ f )(h) represent in this situation?

the ratio of the cost in dollars of applying compost to the cost in dollars of applying fertilizer for a job that takes h hours

The notation (f ÷ g)(x), g(x) ≠ 0 represents the quotient of the functions f (x) and g(x) given that g(x) ≠ 0. In other words, (f ÷ g)(x) = f (x) ÷ g(x), g(x) ≠ 0.

21 Activating Prior Knowledge, Debriefing Note that the Math Tip refers to factoring expressions in the numerator and denominator in Items 21c, 21d, and 21e. Since factoring has not been covered in Algebra 2 at this point, you may wish to review with students the following:

20. Find (c ÷ f )(4). What does this value represent in this situation? 40(4 ) + 140 = 2; For a job that takes 4 hours, the cost of 30(4 ) + 30 applying compost is 2 times the cost of applying fertilizer. ( c ÷ f )(4 ) =

21. Look for and make use of structure. Given f (x) = 2x, g(x) = x + 3, and h(x) = 2x + 6, find each function and simplify the function rule. Note any values that must be excluded from the domain. a. ( f g)(x) (f

(g


2 x = /2 x = x 2 x + 6 /2(x + 3) x + 3

2

⋅ ⋅ h)(x) = (x + 3)(2x + 6) = 2x + 12x + 18

MATH TIP

2

You may be able to simplify the function rules in Items 21c, d, and e by factoring the expression’s numerator and denominator and dividing out common factors.

c. (f ÷ h)(x), h(x) ≠ 0


In Item 21c,

⋅ ⋅ g)(x) = 2x(x + 3) = 2x + 6x

b. (g h)(x)

(f ÷ h)( x ) =

2x = x 2 x + 6 x + 3 , x ≠ −3

18–20 Predict and Confirm, Activating Prior Knowledge Ask students to make a conjecture as to the number of hours (if any) that it would take for the total charge of applying compost to equal the total charge of applying fertilizer. [Students will hopefully realize the impossibility of this because both the hourly charge and material cost are greater for the compost service.]

d. (h ÷ g)(x), g(x) ≠ 0

Discuss that the two in the denominator have a common factor of 2. The coefficient 2 of 2x in the numerator cancels with the common factor of 2 in the denominator. Furthermore, some students are going to want to cancel out the x’s. Be prepared to explain that this is not possible because the x in the denominator is part of the term (x + 3), and the only way to cancel would be if there were a term of (x + 3) in the numerator. In Item 21d,

( h ÷ g )( x ) = 2 x + 6 = 2, x ≠ −3 x+3

2 x + 6 = 2( x + 3 ) = 2 x +3 ( x + 3) In the numerator, there is a common factor of 2 that can be factored out. After doing so, the (x + 3)’s in the numerator and denominator can be entirely canceled.

e. ( g ÷ f )(x) (g ÷ f )( x ) = x + 3 , x ≠ 0 2x

In Item 21e, there is no common factor.



MINI-LESSON: Function Operations If students need additional help with adding, subtracting, multiplying, or dividing functions, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.

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For struggling students, it may be helpful to take some extra time to review factoring out a common factor before moving forward. Here are some suggestions of samples you might use. 1. x2 + 5x = x(x + 5) 2. 10y + 15 = 5(2y + 3) 1 3( p + 4 ) 1 3 p + 12 = = , 3. 6 p + 24 2 2 6( p + 4 ) p ≠ −4


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22. How are operations on functions similar to and different from operations on real numbers?

Sample answer: Operations on real numbers involve only numbers. Operations on functions involve function rules. Otherwise, the processes of addition, subtraction, multiplication, and division are essentially the same. For division of real numbers, the divisor cannot be 0, and for division of functions, the function rule that is the divisor cannot be equal to 0.

With both real numbers and function division, the divisor cannot equal zero. The main differences when performing function operations are the use of function notation and variables. Rather than simple addition, function operations involve combining like . Lastly, function division may require knowledge of factoring polynomials.

Check Your Understanding 23. Given that f (x) = 2x + 1 and g(x) = 3x − 2, what value(s) of x are excluded from the domain of ( f ÷ g)(x)? Explain your answer. 24. Make a conjecture about whether addition of functions is commutative. Give an example that s your conjecture. 25. Given that h(x) = 4x + 5 and (h − j )(x) = x − 2, find j(x). Explain how you determined your answer.

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to function operations.

LESSON 5-1 PRACTICE

Answers

23. x ≠ 2 ; The function (f ÷ g)(x) is 3 undefined when g(x) = 0. Because g(x) = 3x − 2, g(x) = 0 when x = 23 . So, 2 is excluded from the domain 3 of (f ÷ g)(x). 24. Addition of functions is commutative. Sample example: Given that f(x) = 4x + 2 and g(x) = −2x + 5, (f + g)(x) = (g + f)(x) = 2x + 7. 25. j(x) = 3x + 7; Sample explanation: I know that h(x) − j(x) = (h − j)(x), so j(x) = h(x) − (h − j)(x) = 4x + 5 − (x − 2) = 3x + 7.

For Items 26–30, use the following functions. f(x) = 5x + 1

g(x) = 3x − 4

26. ( f + g)(x)

27. ( f − g)(x)

28. ( f g)(x)

29. ( f ÷ g)(x), g(x) ≠ 0

⋅

30. A student incorrectly found (g − f )(x) as follows. What mistake did the student make, and what is the correct answer? (g − f )(x) = 3x − 4 − 5x + 1 = −2x − 3 31. Make sense of problems and persevere in solving them. Jim plans to make a radio ad for his lawn company. The function a(t) = 800 + 84t gives the cost of making the ad and running it t times on an AM station. The function f (t) = 264t gives the cost of running the ad t times on a more popular FM station. a. Find (a + f )(t) and tell what it represents in this situation.

ASSESS

b. Find (a + f )(12) and tell what it represents in this situation.


ADAPT Check students’ answers to the Lesson Practice to ensure that they understand function operations. Sometimes students understand the concepts but are confused by the notation. If this happens, encourage them to begin by rewriting an expression such as (f + g)x as f(x) + g(x) and then substitute expressions for f(x) and g(x).

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LESSON 5-1 PRACTICE

(f + g)(x) = 5x + 1 + 3x − 4 = 8x − 3 (f − g)(x) = 5x + 1 − (3x − 4) = 2x + 5 (f g)(x) = (5x + 1)(3x − 4) = 15x2 − 17x − 4 ( f ÷ g ) = 5x + 1 , x ≠ 43 3x − 4 When subtracting the rule for f(x) from the rule for g(x), the student should have written the rule for f(x) in parentheses so that both of the rule would be subtracted. The correct answer is (g − f)(x) = 3x − 4 − (5x + 1) = −2x − 5. 31. a. (a + f)(t) = 800 + 84t + 264t = 800 + 348t; The function (a + f)(t) represents the cost of making the ad and running it on both stations t times. b. (a + f)(12) = 800 + 348(12) = 4976; It will cost $4976 to make the ad and run it on both stations 12 times.

26. 27. 28. 29. 30.

⋅



Find each function and simplify the function rule. Note any values that must be excluded from the domain.


22 Debriefing This item guides students toward the conclusion that operations with functions follow similar processes and rules as operations with numbers. As with numbers, addition and multiplication of functions follow the commutative properties, whereas subtraction and division do not.




Lesson 5-2 Function Composition


Learning Targets:

• Write functions that describe the relationship between two quantities. • Explore the composition of two functions through a real-world scenario.

PLAN

My Notes

Pacing: 1 class period Chunking the Lesson #1–2 #3–5 #6–8 #9 #10 #11–14 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Create Representations, Identify a Subtask, Group Presentation, Graphic Organizer, Debriefing, Self Revision/Peer Revision

Recall that Jim has a lawn service called Green’s Grass Guaranteed. On every mowing job, Jim charges a fixed $30 fee to cover equipment and travel expenses plus a $20 per hour labor charge. Work with your group on Items 1–14. 1. On a recent mowing job, Jim worked for 6 hours. What was the total charge for this job? $150; 30 + (20 × 6) = 150

2. Model with mathematics. If Jim works for t hours, what will he charge for a mowing job? Write your answer as a cost function where c(t) is Jim’s charge for t hours of work. c(t) = 30 + 20t

TEACH DISCUSSION GROUP TIP With your group, reread the problem scenarios in this lesson as needed. Make notes on the information provided in the problems. Respond to questions about the meaning of key information. Summarize the information needed to create reasonable solutions, and describe the mathematical concepts your group uses to create its solutions.

It takes Jim 4 hours to mow 1 acre. Jim prepares a cost estimate for each customer based on the size (number of acres) of the property.


3. The APCON company is one of Jim’s customers. APCON has 2 acres that need mowing. How many hours does that job take?


Lesson 5-2

8 hours

Bell-Ringer Activity Write the equations y = 6x and y = 6x + 11 on the board. Have students discuss what kind of equations these represent and their similarities and differences. Use the discussion as an opportunity to review independent and dependent variables in equations, and how they relate to domain and range of functions. 1–2 Create Representations, Debriefing Item 1 serves as a concrete, numerical check for student understanding of the scenario. Item 2 asks students to write an algebraic rule relating cost and time. Use Item 2 as a quick assessment of understanding of function notation. 3–5 Quickwrite, Identify a Subtask, Group Presentation Obtaining a cost estimate provides a numerical example for a two-step process that involves composing one function with another. Be sure that all students understand the numerical process before they encounter the symbolic function notation later on. The term composition will be introduced later in the activity.

4. Another customer has a acres of property. Write the equation of a function in of a for the number of hours t it will take Jim to mow the property. t(a) = 4a

5. How much will Jim charge APCON to mow its property? Justify your answer.

$190. Sample explanation: Use the number of hours (8) to substitute into the cost function: c(t) = 30 + 20t = 30 + 20(8) = 190 dollars.


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ACTIVITY 5 Continued 6 Think-Pair-Share Use this item to assess students’ understanding of slope. It will also help them to focus on the quantities represented by each variable.



TEACHER to TEACHER In the linear equation y = mx + b, m is the constant rate of change of y with respect to x. When that equation is graphed, m indicates the slope of the line. Therefore, slope is the graphical representation of rate of change. The rate of change has two representations—a numerical one with units of measure (e.g., $20/h) and a graphical one (e.g., slope of 20). 7–8 Think-Pair-Share, Graphic Organizer, Debriefing Domain and range of the functions are represented in a table and in a graphic organizer. These items set the stage for understanding how the domain and range of a composite function are related to the domain and range of the functions from which it is created.

The functions in Items 2 and 4 relate three quantities that vary, based on the needs of Jim’s customers:

• The size in acres a of the property • The time in hours t needed to perform the work • The cost in dollars c of doing the work. 6. Attend to precision. Complete the table below by writing the rate of change with units and finding the slope of the graph of the function.

MATH TIP In a linear function f (x) = mx + b, the y-intercept is b. The variable m is the rate of change in the values of the function—the change of units of f (x) per change of unit of x. When the function is graphed, the rate of change is interpreted as the slope. So y = mx + b is called the slopeintercept form of a linear equation.

Function

Rate of Change (with units)

Slope

c(t) = 30 + 20t

$20 per hour

20

t(a) = 4a

4 hours per acre

4

7. Complete the table below by naming the measurement units for the domain and range of each function. Function Notation

Description of Function

Domain (units)

Range (units)

c(t)

cost for job

hours

dollars

t(a)

time to mow

acres

hours

8. Calculating the cost to mow a lawn is a two-step process. Complete the graphic organizer below by describing the input and output, including units, for each part of the process.

Output: Time (in hours) Input: Time (in hours)

Cost for Job Output: Cost (in dollars)

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MINI-LESSON: Linear Functions and Slope If students need additional help with identifying the slope and y-intercept of a linear function, as well as interpreting slope and rate of change, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.

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Time to Mow


Input: Area (in acres)






The graphic organizer shows an operation on two functions, called a composition. The function that results from using the output of the first function as the input for the second function is a composite function. In this context, the composite function is formed by the time-to-mow function and the cost-for-job function. Its domain is the input for the time function, and its range is the output from the cost function. 9. Make sense of problems. The cost to mow is a composite function. Describe its input and output as you did in Item 8. Input:

Area (in acres)

MATH A composition is an operation on two functions that forms a new function. To form the new function, the rule for the first function is used as the input for the second function. A composite function is the function that results from the composition of two functions. The range of the first function becomes the domain for the second function.

Area (in acres)

Input:

My Notes

Time to Mow

Number of Hours

Cost (in dollars)

When a composite function is formed, the function is often named to show the functions used to create it. The cost-to-mow function, c(t(a)), is composed of the cost-for-job and the time-to-mow functions.

10 Marking the Text, Graphic Organizer This item allows for a different representation to help students identify the domain and range of the composite function. The representations in Items 9 and 10 parallel the representations in Items 7 and 8.

The c(t(a)) notation implies that a was assigned a value t(a) by the timeto-mow function. Then the resulting t(a) value was assigned a value c(t(a)) by the cost-to-mow function.



10. Complete the table by writing a description for the composite function c(t(a)). Then name the measurement units of the domain and range. Function Notation

Description of Function

c(t(a))

cost to mow

Domain (units)

Range (units)

acres

dollars

The word composition means the act of combining parts to form a whole. Pertaining to math functions, composition is the act of forming a new function by involving two or more functions in succession. It involves taking the output of the first function and using it as the input for the second function, and so on. The result of following these steps creates an entirely new function, called a composite function. It is important to note that when working with composite functions, students start from the inside and work their way outward, similar to order of operations with arithmetic. 9 Think-Pair-Share, Graphic Organizer Students will recognize that composition of the two functions creates a new function, the cost-to-mow function, and will identify the domain and range of the composite function.

Cost for Job

Output:



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ACTIVITY 5 Continued 11–14 Quickwrite, Group Presentation, Debriefing Work with students to analyze what happens in this situation. Students should see that the time function has been substituted into the cost function. The resulting composite is a function of a acres. Students will practice function composition later in this activity.



Jim wants to write one cost function for mowing a acres of property. To write the cost c as a function of a acres of property, he substitutes t(a) into the cost function and simplifies. c(t) = c(t(a)) c(t(a)) = c(4a)

Use Item 11 to determine whether or not students understand the process of composition.

= 30 + 20(4a)

Substitute t(a) for t in the cost function. t(a) = 4a, so write the function in of a. Substitute 4a for t in the original c(t) function.

c(t(a)) = 30 + 80a

Item 12 brings students back to the context. During debriefing, make sure that the point is made that composition is an efficient process if you need to repeatedly use the two-step process represented by composition.

11. Attend to precision. Write a sentence to explain what the expression c(t(2)) represents. Include appropriate units in your explanation. It represents the cost in dollars to mow 2 acres.

Use student communication to reinforce understanding of the meaning of composition. Be sure that students see the different meanings of the value 50 in Items 13 and 14.

12. Construct viable arguments. Why might Jim want a single function to determine the cost of a job when he knows the total number of acres? Sample response: If Jim needs to determine the cost for different-sized properties, this function will be useful.

13. Explain what the expression c(t(50)) represents. Include appropriate units in your explanation.

It represents the number of acres that can be mowed for $50.

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MINI-LESSON: Composition Function Notation If students need additional help with understanding the meaning of composite function notation, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.

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14. Explain what information the equation c(t(a)) = 50 represents. Include appropriate units in your explanation.


It represents the cost in dollars to mow 50 acres.






Check Your Understanding 15. Given the functions a(b) = b + 8 and b(c) = 5c, write the equation for the composite function a(b(c)) and evaluate it for c = −2.

Answers

16. The first function used to form a composite function has a domain of all real numbers and a range of all real numbers greater than 0. What is the domain of the second function in the composite function? Explain.

15. a(b(c)) = 5c + 8; a(b(−2)) = −2 16. All real numbers greater than 0; The domain of the second function in the composite function is the range of the first function. 17. Sample answer: The composite function is composed of the functions f and g. The variable x represents the input. The composite function first evaluates the function g for the value of x to give a value for g(x). Then it evaluates the function f for the value of g(x) to give a value for f(g(x)).

17. The notation f ( g (x)) represents a composite function. Explain what this notation indicates about the composite function.

LESSON 5-2 PRACTICE Model with mathematics. Hannah’s Housekeeping charges a $20 flat fee plus $12 an hour to clean a house. 18. Write a function c(h) for the cost to clean a house for h hours. 19. What are the units of the domain and range of this function? 20. What is the slope of this function? Interpret the slope as a rate of change.

ASSESS

Hannah’s Housekeeping can clean one room every half hour.


21. Write a function h(r) for the hours needed to clean r rooms. 22. Write a function c(h(r)) to represent the cost of cleaning r rooms. 23. What is the value and meaning of c(h(12))? 24. Look for and make use of structure. Explain how a composition of functions forms a new function from the old (original) functions.




Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to composition of functions. Be sure students understand what the domain and range of the composition are in the context of the situation.

LESSON 5-2 PRACTICE

18. c(h) = 20 + 12h 19. domain: hours; range: dollars 20. The slope is 12. It means the cost will increase by $12 for each additional hour of cleaning time. 21. h(r) = 0.5r 22. c(h(r)) = 20 + 12(0.5r) = 20 + 6r 23. $92 is the cost to clean 12 rooms. 24. Possible answer: The output of one function becomes the input of another function.

ADAPT Activity 5 • Function Composition and Operations

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Check students’ answers to the Lesson Practice to ensure that they understand basic concepts related to function composition. If students are confused by the notation, encourage them to write each step. For example, in Item 22, they can begin by writing c(h(r)) = c(0.5r). If students have difficulty substituting algebraic expressions into functions, encourage the use of colored pencils or highlighters. For example, students can write c(h) = 20 + 12h, highlighting both h’s, and then write c(0.5r) and highlight 0.5r. This can help students see what should be substituted into the function (0.5r) and where it should be substituted.


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Lesson 5-3 More Function Composition


PLAN

My Notes

Pacing: 1 class period Chunking the Lesson #1 #2–3 Check Your Understanding #7–11 Check Your Understanding Lesson Practice

Learning Targets:

• Write the composition of two functions. • Evaluate the composition of two functions.

SUGGESTED LEARNING STRATEGIES: Note Taking, Create Representations, Think-Pair-Share, Group Presentation, Debriefing

A composition of functions forms a new function by substituting the output of the inner function into the outer function. The function y = f (g(x)) is a composition of f and g where g is the inner function and f is the outer function.

TEACH

1. The tables show information about Jim’s mowing service. Use the tables to evaluate each expression. Then tell what the expression represents.

Bell-Ringer Activity Have students review (without function notation) substituting values from one expression into another and simplifying, in order to help them with the composite functions. 1. Find 2x + 9, if x = 14. [37] 2. Find 3r − 4 in of x, if r = x − 6. [3x − 22] 3. Find −4a in of b, if a = b + 12. [−4b − 48] Have students discuss their results prior to moving forward with the lesson.

Area of Property a (acres)

Time to Mow t(a) (hours)

Time to Mow t (hours)

Cost to Mow c(t) ($)

1

4

4

110

2

8

8

190

3

12

12

270

4

16

16

350

TEACHER to TEACHER

b. c(4) 110; It costs $110 for 4 hours of mowing.

The main focus of the remainder of this activity is practice with composition of functions. Students compose functions numerically and algebraically. They also learn an alternate notation for composition and work with more typical representations of functions using f, g, and h as functions of x.

c. c(t(1)) 110; It costs $110 to mow 1 acre.

1 Create Representations In Item 1, the use of the two tables, one for each function, provides a visual way of demonstrating how the output, or range, of the first (inner) function becomes the domain of the second (outer) function, as column 2 from the first table is mirrored as column 1 in the second table. The first table represents the t function, and the second table represents the c function.

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d. c(t(4)) 350; It costs $350 to mow 4 acres.

MATH TIP

2. Reason quantitatively. Use the tables of values below to evaluate each expression.

The order matters when you compose two functions. y = g(f (x)) and y = f (g(x)) are two different functions.

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x

f(x)

x

g(x)

1

3

1

4

2

2

2

3

3

1

3

2

4

4

4

1

a. f (3) 1

b. g(3) 2

c. g ( f (3)) 4

d. f (g(3)) 2




a. t(4) 16; It takes 16 hours to mow 4 acres.


Lesson 5-3






3. Using f and g from Item 2, complete each table of values to represent the composite functions ( f g)(x) and ( g f )(x). a.

x

( f g)(x) = f(g(x))

b.

x

(g f )(x) = g(f(x))

My Notes

WRITING MATH

1

4

1

2

The notation (f g)(x) represents a composition of two functions.

2

1

2

3

(f g)(x) = f (g)(x)

3

2

3

4

4

3

4

1

Read the notation as “f of g of x.”

Debrief students’ answers to these items to ensure that they understand concepts related to function compositions.

Answers

4. What is does the notation ( g h)(t) represent? What is another way you can write ( g h)(t)?

⋅

5. Reason abstractly. Explain how (f g)(x) is different from (f g)(x). 6. Given that p(t) = t2 + 4 and q(t) = t + 3, write the equation for ( p q)(t). Explain how you determined your answer.

For Items 7–11, use these three functions:



CONNECT TO AP

f (x) = x2 g (x) = 2x − 1 h(x) = 4x − 3

In AP Calculus, you will identify the “inner” function and the “outer” function that form a composite function.

7. Evaluate each expression. a. g ( f (2)) 7 b. f (g (2)) 9

For example, the function h(x) = f (g(x)) = (2x + 3)2 could be the composition of the inner function g(x) = 2x + 3 and the outer function f (x) = x2.

8. Write each composite function in of x. a. y = g ( f (x)) b. y = f ( g (x)) g(f(x)) = 2x2 − 1

The composite functions that students write in Item 3 will be numeric only, within a table of values. Students are not expected to represent either function or the composite functions algebraically.



• • •

2–3 Create Representations, Predict and Confirm, Debriefing Give students an opportunity to work through Item 2 on their own. If students need help, demonstrate each step in the process of composing the functions numerically.

f(g(x)) = (2x − 1)2 = 4x2 − 4x + 1

9. that you composed g and f correctly by evaluating g ( f (2)) and f ( g (2)) using the functions you wrote in Item 8. Compare your answers with those from Item 7.

4. Sample answer: (g o h)(t) represents a composition of the functions g and h in which h is the inner function and g is the outer function. (g o h)(t) can also be written as g(h(t)). 5. (f o g)(x) is a composition of the functions f and g: (f o g)(x) = f(g(x)). By contrast, (f g)(x) is the product of the functions f and g: (f g)(x) = f(x) g(x). 6. (p o q)(t) = (t + 3)2 + 4 = t2 + 6t + 13; Sample explanation: To find the rule for (p o q)(t), I replaced t in the rule for p(t) with the rule for q(t).

⋅

⋅

⋅

7–11 Think-Pair-Share, Create Representations, Group Presentation, Debriefing These items provide an opportunity for students to compose two functions algebraically. Most of these items use only linear or simple quadratic functions. You may find it necessary to provide further practice with composing different types of functions as they are introduced later in the year. Students should also gain familiarity with the composition notation f o g and g o f.

g(f(2)) = 2(2)2 − 1 = 7

f(g(2)) = (2(2) − 1)2 = 9


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CONNECT TO AP In order to decompose a function, students will need to recognize the inner and outer parts of the composite function in order to correctly apply the chain rule when they study calculus. Examples: Composite f(g(x)) = 5 x − 3

Inner Function g(x) = 5x − 3

Outer Function

Composite f(g(x)) = 4x2 − 1

Inner Function g(x) = x2

Outer Function f(x) = 4x − 1

f (x ) = ( x )


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ACTIVITY 5 Continued Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to evaluating the composition of functions.



10. a. Evaluate h( g (3)). h(g(3)) = h(2(3) − 1) = h(5) = 4(5) − 3 = 17

Answers b. Write the composition (h g)(x) in of x. h(g(x)) = h(2x − 1) = 4(2x − 1) − 3 = 8x − 4 − 3 = 8x − 7

MATH TIP

5; g(g(2)) = g(2(2) − 1) = g(3) = 2(3) − 1 = 5

As shown in Item 11, the inner and outer functions that form a composite function can be the same function.

ASSESS

b. Write the composition ( g g)(x) in of x. (g g)(x) = g(2x − 1) = 2(2x − 1) − 1 = 4x − 3



12. Explain how you found the rule for the composition ( g g)(x) in Item 11b. 13. Given that p(n) = 4n and q(n) = n + 2, for what value of n is (p q)(n) = 8? Explain how you determined your answer.


For Items 14 and 15, use the following functions:

• •

f (x) = 5x + 1 g (x) = 3x − 4

14. Evaluate f (2), g(2), ( f g)(2), and ( g f )(2). 15. Write the composite functions h(x) = g ( f (x)) and k(x) = f ( g (x)). The jeans at a store are on sale for 20% off, and the sales tax rate is 8%. Use this information for Items 16–18. 16. Write a function s(p) that gives the sale price of a pair of jeans regularly priced at p dollars. 17. Write a function t(p) that gives the total cost including tax for a pair of jeans priced at p dollars. 18. Construct viable arguments. A customer wants to buy a pair of jeans regularly priced at $25. Does it matter whether the sales clerk applies the sale discount first or adds on the sales tax first to find the total cost? Use compositions of the functions s and t to your answer.

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ADAPT Check students’ answers to the Lesson Practice to ensure that they understand how to write and evaluate function compositions. Students who make errors in evaluation may be evaluating the functions in the wrong order. Have them write the composition without using the notation o, if necessary, and remind them to begin in the innermost parentheses and work outward.

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LESSON 5-3 PRACTICE

LESSON 5-3 PRACTICE

14. f(2) = 11; g(2) = 2; f(g(2)) = 11; g(f(2)) = 29 15. h(x) = g(f(x)) = 3(5x + 1) − 4 = 15x − 1; k(x) = f(g(x)) = 5(3x − 4) + 1 = 15x − 19 16. s(p) = 0.8p 17. t(p) = 1.08p 18. No. Sample explanation: The composite function s(t(p)) = 0.8(1.08p) = 0.864p gives the total cost of the jeans if the sales tax is added on before the discount is applied. The composite function t(s(p)) = 1.08(0.8p) = 0.864p gives the total cost of the jeans if the discount is applied before the sales tax is added on. Both composite functions have the same rule, so both give the same total cost for the jeans: 0.864($25) = $21.60.

11. a. Evaluate g ( g (2)).


12. Sample answer: I took the rule for g, 2x − 1, and substituted it for x in the rule for g: 2(2x − 1) − 1. Then I simplified the resulting expression: 2(2x − 1) −1 = 4x − 3. 13. n = 0; Sample explanation: First I wrote the rule for (p o q)(n): (p o q)(n) = 4(n + 2) = 4n + 8. Then I substituted 8 for (p o q)(n) and solved for n. (p o q)(n) = 4n + 8 8 = 4n + 8 0 = 4n 0=n




Function Composition and Operations New from Old


Lesson 5-1 Use f (x) = 5x + 2, g (x) = 3 − x, and h(x) = x − 3 to answer Items 1–8. Find each function and simplify the function rule. Note any values that must be excluded from the domain. 1. ( f + g)(x)

2. (h + g)(x) 4. (h − f )(x)

5. ( f g)(x)

6. ( g h)(x)

⋅

⋅

7. ( f ÷ g)(x), g(x) ≠ 0

9. A rectangular skate park is 60 yards long and 50 yards wide. Plans call for increasing both the length and the width of the park by x yards.

Skate Park



x yd

a. Write a function l (x) that gives the new length of the skate park in of x. b. Write a function w(x) that gives the new width of the skate park in of x. c. What does (l w)(x) represent in this situation? Write and simplify the equation for (l w)(x). d. Find (l w)(5), and tell what it represents in this situation.

⋅

Jim wants to calculate the cost of running his lawn mowers. The mowers consume 2.5 gallons of gasoline each hour. Gasoline costs $3.50 per gallon. 13. Write a function g (h) that gives the number of gallons g that the mowers will use in h hours. Identify the units of the domain and range. 14. Write a function c ( g) for the cost c in dollars for g gallons of gasoline. Identify the units of the domain and range.

x yd

60 yd

12. The cost in dollars of renting a car for d days is given by c(d ) = 22d + 25. The cost in dollars of renting a hotel room for d days is given by h(d) = 74d. a. What does (c + h)(d ) represent in this situation? Write and simplify the equation for (c + h)(d). b. For what value of d is (c + h)(d ) = 600? What does this value of d represent in this situation?

Lesson 5-2

8. ( g ÷ h)(x), h(x) ≠ 0

50 yd

continued

11. Make a conjecture about whether subtraction of functions is commutative. Give an example that s your answer.

ACTIVITY 5 PRACTICE

3. ( f − g )(x)

ACTIVITY 5

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⋅

15. Use composition of functions to create a function for the cost c in dollars of gasoline to mow h hours. Identify the units of the domain and range. Then explain how the domain and range of the composite function are related to the domain and range of g (h) and c (g). 16. Use the composite function in Item 15 to determine the cost of gasoline to mow 12 hours. Show your work. 17. What is the slope of the composite function, and what does it represent in this situation?

10. Given that p(n) = 4n2 + 4n − 6 and q(n) = n2 − 5n + 8, find (p − q)(3). A. 26 B. 38 C. 40 D. 42


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ACTIVITY PRACTICE 1. (f + g)(x) = 5x + 2 + (3 − x) = 4x + 5 2. (h + g)(x) = x − 3 + 3 − x = 0 3. (f − g)(x) = 5x + 2 − (3 − x) = 6x − 1 4. (h − f)(x) = x − 3 − (5x + 2) = −4x − 5 5. (f g)(x) = (5x + 2)(3 − x) = −5x2 + 13x + 6 6. (g h)(x) = (3 − x)(x − 3) = −x2 + 6x − 9 7. ( f ÷ g )(x ) = 5x + 2 , x ≠ 3 3− x x = −1, x ≠ 3 8. ( g ÷ h)(x ) = 3x − −3

⋅ ⋅

9. a. l(x) = x + 60, where l(x) is the new length in yards and x is the increase in length in yards b. w(x) = x + 50, where w(x) is the new width in yards and x is the increase in width in yards c. the new area of the skate park in square yards; (l w)(x) = (x + 60)(x + 50) = x2 + 110x + 3000 d. (l w)(5) = 3575; The new area of the skate park will be 3575 square yards if its length and width are each increased by 5 yards. 10. C 11. Subtraction of functions is not commutative. Sample example: Given that f(x) = 2x + 6 and g(x) = 4x + 4, (f − g)(x) = −2x + 2 and (g − f)(x) = 2x − 2. So, (f − g)(x) ≠ (g − f)(x). 12. a. the cost in dollars of renting both a car and a hotel room for d days; (c + h)(d) = 22d + 25 + 74d = 96d + 25 b. d ≈ 5.99; For $600, you can rent both a car and a hotel room for about 6 days. 13. g(h) = 2.5h; domain: hours, range: gallons 14. c(g) = 3.5g; domain: gallons, range: dollars 15. c(g(h)) = 3.5(2.5h) = 8.75h; domain: hours, range: dollars. The domain of this new function is the inner function’s domain. The range is the outer function’s range. 16. c(g(12)) = 8.75(12) = 105. It will cost $105 to run the lawn mowers for 12 hours. 17. 8.75; the cost in dollars of running the lawn mowers for 1 hour

⋅

⋅


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ADDITIONAL PRACTICE If students need more practice on the concepts in this activity, see the Teacher Resources for additional practice problems.

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Function Composition and Operations New from Old

An empty swimming pool is shaped like a rectangular prism with a length of 18 feet and a width of 9 feet. Once water begins to be pumped into the pool, the depth of the water increases at a rate of 0.5 foot per hour. 18. Write a function d(t) that gives the depth in feet of the water in the pool after t hours. 19. Write a function v (d) that gives the volume in cubic feet of the water in the pool when the depth of the water is d feet. 20. Write the equation of the composite function v(d(t)), and tell what this function represents in this situation. 21. What is v(d(4)), and what does it represent in this situation? 22. The range of the function d(t) is 0 ≤ d ≤ 4. Based on this information, what is the greatest volume of water the pool can hold?

Lesson 5-3 Use f (x) = x2, g (x) = x + 5, and h(x) = 4x − 6 to answer Items 23–28. Find each function and simplify the function rule. 23. ( f g )(x)

24. (g f )(x)

25. ( f h )(x)

26. ( h f )(x)

27. ( g h )(x)

28. ( h g )(x) The function c( f ) = 5 ( f − 32) converts a temperature 9 f in degrees Fahrenheit to degrees Celsius. The function k(c) = c + 273 converts a temperature c in degrees Celsius to units called kelvins. 29. Write a composite function that can be used to convert a temperature in degrees Fahrenheit to kelvins.

31. Given that (r s)( t ) = 2t + 11, which could be the functions r and s? A. r (t) = t + 1, s(t) = 2t + 5 B. r (t) = t + 5, s(t) = 2t + 1 C. r (t) = 2t + 1, s(t) = t + 5 D. r (t) = 2t + 5, s(t) = t + 1 32. What is the composition f g if f (x) = 4 − 2x and g (x) = 3x2? A. f ( g (x)) = 12x2 − 6x3 B. f ( g (x)) = 4 − 6x2 C. f ( g (x)) = 3(4 − 2x)2 D. f ( g (x)) = 12 − 12x4 Use f (x) = 5x + 2 and g (x) = 3 − x to answer Items 33–35. 33. What is the value of f ( g (−1)) and g ( f (−1))? 34. What is the composite function y = f ( g (x))? 35. What is the composite function y = g( f (x))?

MATHEMATICAL PRACTICES Model with Mathematics

36. A store is discounting all of its television sets by $50 for an after-Thanksgiving sale. The sales tax rate is 7.5%. a. Write a function s(p) that gives the sale price of a television regularly priced at p dollars. b. Write a function t(p) that gives the total cost including tax for a television priced at p dollars. c. A customer wants to buy a television regularly priced at $800. Does it matter whether the sales clerk applies the sale discount first or adds on the sales tax first to find the total cost? Use compositions of the functions s and t to your answer.

30. In Item 29, does it matter whether you wrote (c k)( f ) or (k c)( f )? Explain.

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18. d(t) = 0.5t 19. v(d) = 162d 20. v(d(t)) = 162(0.5t) = 81t; the volume in cubic feet of water in the pool after t hours 21. v(d(4)) = 324; After 4 hours, the volume of water in the pool will be 324 cubic feet. 22. 648 cubic feet 23. (f o g)(x) = (x + 5)2 = x2 + 10x + 25 24. (g o f)(x) = x2 + 5 25. (f o h)(x) = (4x − 6)2 = 16x2 − 48x + 36 26. (h o f)(x) = 4x2 − 6 27. (g o h)(x) = 4x − 6 + 5 = 4x − 1 28. (h o g)(x) = 4(x + 5) − 6 = 4x + 14 29. (k o c)( f ) = 5 ( f − 32) + 273 9 30. Yes. Sample explanation: The rule for (k o c)( f ) simplifies to 5 f + 2297 , and the rule for 9 9 (c o k)( f ) simplifies to 5 f + 1205. 9 9 These two functions are not the same, so it matters which one you use. 31. C 32. B 33. f(g(−1)) = 22, g(f(−1)) = 6 34. f(g(x)) = 17 − 5x 35. g(f(x)) = 1 − 5x 36. a. s(p) = p − 50 b. t(p) = 1.075p c. Yes. Sample explanation: The composite function s(t(p)) = 1.075p − 50 gives the total cost of a television set if the sales tax is added on before the discount is applied. This rule gives a total cost of $810 for the television set regularly priced at $800. The composite function t(s(p)) = 1.075(p − 50) = 1.075p − 53.75 gives the total cost of a television set if the discount is applied before the sales tax is added on. This rule gives a total cost of $806.25 for the television set regularly priced at $800.




ACTIVITY

Inverse Functions

ACTIVITY 6

Old from New Lesson 6-1 Finding Inverse Functions Learning Targets:

• Find the inverse of a function. • Write the inverse using the proper notation.

Activity Standards Focus My Notes

Lesson 6-1

Green’s Grass Guaranteed charges businesses a flat fee of $30 plus $80 per acre for lawn mowing. For residential customers who may have a more limited budget, Jim Green needs to determine the size of the yard he could mow for a particular weekly fee.

PLAN

Work on Items 1–10 with your group.

3 of an acre 8

b. $80 5 of an acre 8

DISCUSSION GROUP TIP As you work in groups, review the problem scenario carefully and explore together the information provided and how to use it to create a potential solution. Discuss your understanding of the problem and ask peers or your teacher to clarify any areas that are not clear.

c. $110


1 acre


Guided In Activity 6, students find inverse functions. They also use composition of functions to determine if functions are inverses of one another. Throughout this activity, emphasize how the domain of a function is related to the range of the inverse of the function and vice versa.

SUGGESTED LEARNING STRATEGIES: Questioning the Text, Think-Pair-Share, Work Backward, Debriefing, Quickwrite, Create Representations, Look for a Pattern, Group Presentation, Note Taking

1. The cost function F is C = F(A). It can be written C = 30 + 80A, where C is the cost to mow A acres. Use the function to determine what part of an acre Jim could mow for each weekly fee. a. $60

6

Pacing: 1 class period Chunking the Lesson #1

#2–6

#7–10

Example A

Check Your Understanding Lesson Practice

TEACH Bell-Ringer Activity As preparation for finding inverses of functions, students should be fluent in solving literal equations. Have students solve each equation for the indicated variable: 1. y = 5x + 8 for x in of y  y −8 x =  5   2. 3a + 2b = 7 for a in of b  a = 7 − 2b  3   3 3. c − 9d = 12 for c in of d 4 [c = 12d + 16]

To make a profit and still charge a fair price, Jim needs a function for calculating the maximum acreage that he can mow, based on the amount of money a customer is willing to spend. 2. Attend to precision. What are the units of the domain and range of the cost function in Item 1? domain: acres, range: dollars

1 Think-Pair-Share, Work Backward Item 1 gives students an opportunity to solve the cost equation repeatedly for a given dollar amount. Students use this process later to write a rule for A acres as a function of C dollars.

3. Make use of structure. Solve the function equation from Item 1 for A in of C. C = 30 + 80A A = C − 30 80

4. Write the answer equation from Item 3 using function notation, where G is the acreage function. G(C ) = C − 30 80

Activity 6 • Inverse Functions

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Common Core State Standards for Activity 6 HSF-BF.B.4

Find inverse functions.

HSF-BF.B.4a

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse.

HSF-BF.B.4b(+) by composition that one function is the inverse of another. [Note: HSF-BF.B.4b is introduced in this activity but is also addressed in higher level mathematics courses.]

2–6 Work Backward, Quickwrite, Debriefing Item 2 sets the stage for learning about the properties of inverse functions, as students will eventually learn that the domain of a function is the range of the inverse of the function and the range of a function is the domain of the inverse of the function. In Items 3 and 4, students form the inverse of the cost function by solving for A. However, the math concept and vocabulary term inverse function are not used at this point in the activity.


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Lesson 6-1 Finding Inverse Functions


5. What are the units of the domain and range of the function G? domain: dollars, range: acres

6. Reason abstractly. How are the domain and range of F(A) related to those of G(C)? Use your response to Item 3 to explain why the relationship exists. The domain of the cost function C = F(A) is the range of the acreage function G. The range of C is the domain of G. Sample explanation: The function C was rewritten to express A in of C, thereby forming the function G.

7. Use the appropriate functions to evaluate each expression. a. G(60) 3 8

b. F(G(60))


60

Discuss that a function relates each element of a set with exactly one element of another set. The domain is the set of values that can go into a function, which are sometimes referred to as “input values.” The range is the set of values that actually come out of a function, which are sometimes referred to as “output values.”

c. F(2) 190

d. G(F(2)) 2

8. Attend to precision. Interpret the meaning of each expression and its corresponding value in Item 7. Be sure to include units in your explanation. a. In part a, you can mow 38 of an acre for $60. b. In part b, it costs $60 to mow the number of acres you can mow for $60.

c. In part c, it will cost $190 to mow 2 acres. d. In part d, you can mow 2 acres for the cost of mowing 2 acres.

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2–6 (continued) In Items 5 and 6, students are asked to name the domain and range of the function G and then to identify the relationship between the domains and ranges of the two functions. The domains and ranges switch in these two functions, meaning that the domain of F is the range of G and that the domain of G is the range of F. Students are first asked to recognize this relationship prior to determining that these are inverse functions. Later, they will discover that the operations performed by the inner function in each composition are then undone by the operations performed by the outer function. This is the core concept of inverse functions that all students must understand after finishing this activity.








9. In general, what is the result when you evaluate F(G(x)) and G(F(x))? You get x.

10. What do the answers in Items 7–9 suggest about F and G? Sample answer: They undo each other.

Two functions f and g are inverse functions if and only if: f(g(x)) = x for all x in the domain of g, and g( f(x)) = x for all x in the domain of f.

MATH Functions f and g are inverse functions if and only if f(g(x)) = x for all x in the domain of g and g(f(x)) = x for all x in the domain of f.

The function notation f –1 denotes the inverse of function f and is read “f inverse.” Item 6 showed that the domain of a function is the range of its inverse. Likewise, the range of a function is the domain of its inverse. To find the inverse of a function algebraically, interchange the x and y variables and then solve for y.

WRITING MATH

Example A



Find the inverse of the function f(x) = 2x − 4. Step 1: Let y = f(x). Step 2: Interchange the x and y variables. Step 3: Solve for y.

Let y = f −1(x). Solution: f −1 (x ) = x + 4 2 Step 4:

f −1( x ) = −x + 8 3

c. h(x) = 2 x − 5 3

h−1( x ) = 3 x + 15 2

Example A Note Taking Work through this example with your students step-by-step to demonstrate how to find the inverse of a function algebraically. Make sure to connect back to the work they did on the previous two pages of this activity, relating the domain and range of inverse functions.

If f and g are inverse functions, you can also write two equivalent composite functions:

y = 2x − 4 x = 2y − 4 x + 4 = 2y y = x+4 2 x + 4 −1 f (x ) = 2

f g=x g f=x

MATH TIP The –1 superscript in the function notation f −1 is not an exponent, and f −1 ≠ 1 when referring to f functions.

Try These A Find the inverse of each function. a. f(x) = −3x + 8

7–10 Create Representations, Quickwrite, Think-Pair-Share, Debriefing Students need to interpret the meaning of F(G(60)) = 60 and G(F(2)) = 2 to understand the effect that composition of these functions has upon domain values. These functions undo each other, and, once again, this key concept of inverse functions is presented without actually mentioning the term inverse functions. While the verbal explanations of the composition may seem self-evident, they emphasize that the functions are, in fact, undoing each other. Item 8 asks students to attend to precision by using units to describe the answers.

b. g (x ) = 1 (x + 12) 4

However, for any number n, the expression n–1 is the multiplicative inverse, or reciprocal, of n.

g–1(x) = 4x − 12

d. j(x ) = 3x − 2 6

j −1( x ) = 2 x + 2 3


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MINI-LESSON: Inverse vs. Reciprocal Notation If students need additional help understanding the notation of f−1, a mini-lesson is available to provide practice. See the Teacher Resources at SpringBoard Digital for a student page for this mini-lesson.


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ACTIVITY 6 Continued continued My Notes


Answers 11. Because f and g are inverse functions, they undo each other, which means that f(g(x)) = x; f(g(20)) = 20. 12. No. Sample justification: If h(x) and j(x) are inverse functions, then h(j(x)) should equal x. Instead, h(j(x)) = 3(−3x) = −9x. Therefore, h(x) and j(x) are not inverse functions. 13. Domain: x ≤ −2, range: y ≥ 3; The domain of a function is the range of its inverse, and the range of a function is the domain of its inverse. 14. a. D(T ) = 6 T = 1 T 60 10 b. domain: minutes, range: miles; domain: [0, ∞), {T | T ∈ , T ≥ 0}, range: [0, ∞), {D | D ∈ , D ≥ 0} c. domain: miles, range: minutes; domain: [0, ∞), {D | D ∈ , D ≥ 0}, range: [0, ∞), {T | T ∈ , T ≥ 0} d. 25 minutes; Sample explanation: Let T(D) represent the inverse of D(T). Because D(T ) = 1 T, 10 T gives the number of miles Mariana can run in T minutes; T(D) = 10D gives the number of minutes it takes Mariana to run D miles. Evaluate T(D) for D = 2.5: T(2.5) = 10(2.5) = 25.

11. Given that f and g are inverse functions, explain how you can find f (g(20)) without knowing the equations for f and g. 12. Critique the reasoning of others. A student claims that h(x) = 3x and j(x) = −3x are inverse functions. Is the student correct? Justify your answer. 13. The domain of a function is x ≥ 3, and the range of the function is y ≤ −2. What are the domain and range of the inverse function? Explain your answer. 14. Mariana’s average running speed is 6 miles per hour. a. Write a function D(T) that gives the distance in miles Mariana covers when running for T minutes. b. What are the units of the domain and range of D(T)? Write the domain and range in both interval notation and set notation.

( )

c. What are the units of the domain and range of the inverse of D(T)? Write the domain and range of the inverse function in both interval notation and set notation. d. How many minutes will it take Mariana to run 2.5 miles? Explain how you can find the answer by using the inverse of D(T).

LESSON 6-1 PRACTICE Look for and make use of structure. The function T = F(H) estimates the temperature (degrees Celsius) on a mountain given the height (in meters) above sea level. Use the function T = 50 − H . 20 15. What is F(500)? What does F(500) mean? 16. Find H in of T. Label this function G. 17. What is G(25)? What does G(25) mean? 18. Are the functions F and G inverses of each other? Explain.

Find the inverse of each function. 19. f (x) = 3x + 6

ASSESS

21. h(x ) = x − 20 4


20. g (x ) = − 1 x 2 22. j(x) = 5(x − 1)


ADAPT Check students’ answers to the Lesson Practice to ensure that they understand inverse functions. If students struggle with the concept of inverse functions, relate it to inverse operations. Just as inverse operations “undo” each other, so do inverse functions. Demonstrate by evaluating a function for a given value, and then substituting the result into the inverse function; the answer is the original value. The inverse function has “undone” the original function.

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LESSON 6-1 PRACTICE

15. 25; When the elevation is 500 m, the temperature is 25 degrees Celsius. 16. G(T) = H = 1000 − 20T 17. 500; When the temperature is 25 degrees Celsius, the elevation is 500 m. 18. Yes. G(F(x)) = F(G(x)) = x 19. f −1 (x ) = 1 x − 2 3 20. g−1(x) = −2x 21. h−1(x) = 4x + 20 22. j−1 (x ) = 1 x + 1 5



Debrief students’ answers to these items to ensure that they understand concepts related to inverse functions.


ACTIVITY 6






Lesson 6-2 Graphs of Inverse Functions


Lesson 6-2 PLAN

Learning Targets:

composition of functions to determine if functions are inverses of • Use each other. • Graph inverse functions and identify the symmetry.

My Notes

Pacing: 1 class period Chunking the Lesson Example A Check Your Understanding #4–6 #7–8 Check Your Understanding Lesson Practice

SUGGESTED LEARNING STRATEGIES: Think-Pair-Share, Create Representations, Group Presentation, Quickwrite, Debriefing, Discussion Groups, Self Revision/Peer Revision You can use the definition of inverse functions to show that two functions are inverses of each other.

ACADEMIC VOCABULARY When you prove a statement, you use logical reasoning to show that it is true.

Example A

Use the definition of inverse functions to prove that f(x) = 2x − 4 x+4 −1 and f (x ) = are inverse functions. 2 Step 1: Compose f and f −1. f(f −1(x)) = 2( f −1(x)) − 4 Step 2: Substitute f −1 into f. Step 3: Simplify. Step 4: Compose f −1 and f.



Step 5: Substitute f into f −1.

= 2( x + 4 ) − 4 2 =x+4−4 =x f (x ) + 4 −1 f (f(x)) = 2 = 2x − 4 + 4 2

TEACH Bell-Ringer Activity Have students find the inverse of each function.  f −1 ( x ) = x − 2  1. f(x) = 3x + 2  3   f −1 ( x ) = x + 7  2. f(x) = 5x − 7  5   −1 1 3. f ( x ) = x + 4 [f ( x ) = 2(x − 4)] 2 Example A Note Taking, Debriefing Emphasize that both compositions are necessary to that two functions are inverses.

Try These A Answers

a. f −1 (x ) = x + 14 4 f −1 ( f (x )) = 4 x − 14 + 14 = x 4 −1 x + 14 − 14 = x f ( f (x )) = 4 4 b. g−1(x) = 2x − 6 g ( g −1 (x )) = 1 (2 x − 6) + 3 = x 2 g −1 ( g (x )) = 2 1 x + 3 − 6 = x 2

= 2x Step 6: Simplify. 2 =x Solution: f(x) = 2x − 4 and f −1 (x ) = x + 4 are inverse functions. 2

Try These A Make use of structure. Find the inverse of the function. Then use the definition to prove the functions are inverses. Show your work. a. f(x) = 4x − 14 b. g (x ) = 1 x + 3 2


(

)

(

)

93


93



ACTIVITY 6 Continued continued My Notes


Answers

1. Suppose that the domain of f(x) = 2x − 4 in Example A was restricted to {x | x ∈ R, x ≥ 2}. What would be the domain and range in set notation of f −1(x)? Explain your answer.

1. Domain: {x | x ∈ , x ≥ 0}, range: {y | y ∈ , y ≥ 2}; Sample explanation: If the domain of f(x) = 2x − 4 is {x | x ∈ , x ≥ 2}, then its range is {y | y ∈ , y ≥ 0}. The elements in the domain of f(x) are the elements of the range of f −1(x), and the elements of the range of f(x) are the elements of the domain of f −1(x). 2. Find h(j(x)) and j(h(x)). If both compositions are equal to x, then h(x) and j(x) are inverse functions. 3. No. If q(t) were the inverse of p(t), then the domain of p(t) would be the range of q(t). Because the domain of p(t) is not the same as the range of q(t), the functions are not inverses.

2. Construct viable arguments. Explain how to prove that two functions h(x) and j(x) are inverse functions. 3. The domain of p(t) is [0, ∞). The range of q(t) is (−∞, 0]. Based on this information, could q(t) be the inverse of p(t)? Explain your answer.

You can use the relationship between the domain and range of a function and its inverse to graph the inverse of a function. If (x, y) is a point on the graph of a given function, then (y, x) is a point on the graph of its inverse. 4. Complete the table of values for f(x) = 3x − 2. Use the values to graph the function on the coordinate axes below.

4–6 Think-Pair-Share, Create Representations, Debriefing Students should recognize that switching the columns from Item 4 provides a table for the inverse function in Item 5 because the (x, y) values are reversed. Students who find the inverse algebraically and then use that function to complete the table may not fully understand this property of inverse functions. If students are not simply exchanging the ordered pairs from the first table when they graph the inverse, bring this point out during debriefing.

f(x)

−2

−8

0

−2

6

3

7

4

4

10

2

10

f(x) = 3x – 2

8

Graph shows answers for Items 4–6.

–10 –8 –6 –4

–2

–2

f –1(x) = 2

4

6

(x + 2) 3 x 8 10

–4 –6 –8 –10

5. Use the table in Item 4 to make a table of values for the inverse of f. Then graph the inverse on the same coordinate axes. See Item 4 for graph.

94

94

y

x

x

f −1(x)

−8

−2

−2

0

7

3

10

4




Debrief students’ answers to these items to ensure that they understand concepts related to proving two functions are inverse relations.


ACTIVITY 6








6. Show the graph of y = x as a dotted line on the coordinate axes in Item 4. Describe any symmetry among the three graphs. See Item 4 for graph. The two lines are symmetric about the line y = x.

7. Find the inverse of f(x) = x − 4. f −1(x) = x + 4

My Notes CONNECT TO GEOMETRY Geometric figures in the coordinate plane can have symmetry about a point, a line, or both.

8. a. Model with mathematics. Graph f(x) = x − 4, its inverse f −1(x) from Item 7, and the dotted line y = x on the coordinate axes. y

f –1(x) = x + 4

10

6

Some students may have misconceptions about finding inverses. For example, students may simply use inverse operations. If students state the inverse of f(x) = 2x + 1 as f −1 (x ) = x − 1, work 2 through additional examples as needed to reinforce the process. Inverse functions will be readdressed throughout Algebra 2 and will be in concepts in precalculus and calculus.

4 f(x) = x – 4

2 –2

–2

2

4

6

8

10

x

–4 –6 –8 –10

b. Describe any symmetry that you see on the graph in Item 8a.


The lines are symmetric about the line y = x, but they do not intersect because they are parallel.


As an algebraic justification, students may find the inverse of any line with a slope of 1. In general, if f (x) = x + b, they will find that f −1(x) = x − b.

TEACHER to TEACHER

8

–10 –8 –6 –4

7–8 Create Representations, Debriefing In these items students will see that not all inverses intersect along the line y = x. Ask students to generalize as to which functions intersect the line. They should understand that parallel lines don’t intersect; therefore, any line with a slope of 1 will have an inverse parallel to the line y = x.

Check Your Understanding Debrief students’ answers to these items to ensure that they understand concepts related to visual representations of inverse functions.

Check Your Understanding 9. Graph the function f(x) = 2 and its inverse on the same coordinate plane. Is the inverse of f(x) = 2 a function? Explain your answer. 10. What is the relationship between the slope of a nonhorizontal linear function and the slope of its inverse function? Explain your reasoning. 11. What is the relationship between the x- and y-intercepts of a function and the x- and y-intercepts of its inverse? Explain your reasoning.

MATH TIP Recall that the slope of a linear y − y1 function is equal to 2 , where x 2 − x1 (x1, y1) and (x2, y2) are two points on the function’s graph.

Answers 9. No. The inverse is the vertical line x = 2, which is not a function.

y

inverse of f(x) = 2

4 f(x) = 2

2 2

–2

4

x

–2

11. The x-intercepts of a function are the same as the y-intercepts of its inverse. The y-intercept of a function is the same as the x-intercept of its inverse. Sample explanation: If c is an x-intercept of a function, then the graph of the function includes the point (c, 0) and the graph of the inverse includes the point (0, c). So, c is a y-intercept of the inverse of the function. Likewise, if b is the y-intercept of a function, then the graph of the function includes the point (0, b) and the graph of the inverse includes the point (b, 0). So, b is the x-intercept of the inverse of the function.


95

10. The slopes are reciprocals or multiplicative inverses. Sample explanation: If (x1, y1) and (x2, y2) are 2 points on the graph of a nonhorizontal linear function, then y − y1 the slope of the function is 2 . x2 − x1 The graph of the inverse of the function includes the points (y1, x1) and (y2, x2), so the slope of the x − x1 inverse function is 2 . The y2 − y1 product of the slope of the function and the slope of its inverse is yy22− −yy11 xx22− −xx11 = =11. The slopes xx22− −xx11 yy22− −yy11 must be reciprocals of each other.

⋅


95




ACTIVITY 6

ASSESS

continued


My Notes

12. f(x) = 6 − 3x


13. g(x) = x + 2 14. h(x) = −x + 5 15. Express regularity in repeated reasoning. Using your results in Items 12–14, state whether each statement is true or false. Explain your reasoning. a. A function and its inverse always intersect.

LESSON 6-2 PRACTICE

12. f −1 (x ) = x − 6 , −3 f −1 ( f ( x )) = 6 − 3x − 6 = x , −3 −1 f ( f ( x )) = 6 − 3 x − 6 = x −3

b. The rule for a function cannot equal the rule for its inverse.

In Items 16 and 17, graph the inverse of each function shown on the coordinate plane. y y 16. 17.

)

13. g −1(x) = x − 2, g −1(g(x)) = x + 2 − 2 = x, g(g −1(x)) = x − 2 + 2 = x 14. h −1(x) = −x + 5, h −1(h(x)) = −(−x + 5) + 5 = x, h(h −1(x)) = −(−x + 5) + 5 = x 15. a. False; A function and its inverse may or may not intersect. Item 13 provides a counterexample because the two lines are parallel. b. False; The rule for a function can be the same as the rule for its inverse, as shown in Item 14. y 16.

–4

4

4

2

2 2

–2

4

x

–4

–4

4

2

4

x

–2 –4 inverse

y

17. 4 2 –4

–2 –2

x 96


inverse 18. Sample answer: The domain and range of a function and those of its –4 inverse are interchanged. The graphs of a function and its inverse are symmetric about the line y = x. ADAPT The composition of a function and Check students’ answers to the Lesson its inverse equals x. Practice to ensure that they understand the properties of inverse functions and their graphs. Use reflective devices or paper-folding techniques to students who have difficulty recognizing symmetry about y = x or reflecting graphs across y = x.

96

x


2

4

18. Reason abstractly and quantitatively. Summarize the relationship between a function and its inverse by listing at least three statements that must be true if two functions are inverses of each other.

2 –2

2

–2 –2

4

–4

–4

–2



(

LESSON 6-2 PRACTICE In Items 12–14, find the inverse of each function. Use the definition of inverse functions to that the two functions are inverses.



Inverse Functions Old from New


The function m = F(a) = a gives the distance 8 in inches on a map between two points that are actually a miles apart. Use this function for Items 10–13. 10. What is F(50)? What does F(50) represent? 11. a. What is the inverse of F(a)? Label this function G, and tell how you determined the rule for the inverse function. b. Tell what the inverse function represents. 12. What is G(3)? What does G(3) represent? 13. Two towns on the map are 4 1 inches apart. 2 What is the actual distance in miles between the two towns? Explain how you determined your answer. 14. What is the inverse of the function p(t) = 6t + 8? A. p−1(t) = −6t − 8 −t + 8 B. p−1 (t ) = 6 C. p−1 (t ) = t − 8 6 D. p−1 (t ) = 1 t − 8 6

ACTIVITY 6 PRACTICE


Lesson 6-1 Mark’s landscaping business Mowing Madness uses the function c = F(a) to find the cost c of mowing a acres of land. He charges a $50 fee plus $60 per acre. Mark’s cost-calculating function is c = 60a + 50. Use this function for Items 1–7. 1. What is F(40)? What does F(40) mean? 2. Find a in of c. Label this function G. 3. What is G(170)? What does G(170) mean? 4. a. What are the units of the domain and range of F(a)? b. What are the units of the domain and range of G(c)? 5. a. What are the domain and range of F(a) in interval notation? b. What are the domain and range of G(c) in interval notation? 6. Are F(a) and G(c) inverse functions? Explain your answer.



7. A customer has $200 to spend on mowing. How many acres will Mark mow for this amount? Explain how you determined your answer.

Lesson 6-2 15. Use the definition of inverse to determine whether or not each pair of functions are inverses. a. f(x) = 5x − 3, g (x ) = x + 3 5 b. f ( x ) = x + 3, g(x) = 2x − 6 2 c. f(x) = 2(x − 4), g (x ) = 1 x + 4 2 d. f(x) = x + 3, g(x) = −x − 3

8. Find the inverse of each function. a. f(x) = 2x − 10 b. g (x ) = x + 5 4 c. h(x ) = 1 (x − 8) 6 d. j(x) = −5x + 2 9. Given that f(1) = 5, which of the following statements must be true? A. f −1(1) = −5 B. f −1(1) = 5 D. f −1(5) = 1 C. f −1(5) = −1

16. f and h are inverses because they are symmetric about the line y = x. y

20

8 4

–4

f

−1

4

(x) −2

−1

2

0

5

0

1

3

5

(

12

–8

x

18. a. f −1 (x ) = − 1 x + 1, 3 −1 f ( f ( x )) = −3 − 1 x + 1 + 3 = x, 3

16

–8 –4


17.

4

8

12 16 20

x

)

f −1 ( f (x )) = − 1 (−3x + 3) + 1 = x 3 b. g −1(x) = 4x − 2.4, g(g −1(x)) = 0.25(4x − 2.4) + 0.6 = x, g −1(g(x)) = 4(0.25x + 0.6) − 2.4 = x

97

ACTIVITY PRACTICE 1. F(40) = 60(40) + 50 = 2450; F(40) is the cost in dollars of mowing 40 acres. 2. a = c − 50 ; G(c) = c − 50 60 60 170 − 50 3. G(170) = = 2; G(170) is 60 the number of acres that Mark will mow for $170. 4. a. domain: acres; range: dollars b. domain: dollars; range: acres 5. a. domain: (0, ∞); range: (50, ∞) b. domain: (50, ∞); range: (0, ∞) 6. Yes. Sample explanation: The functions are inverses because F (G(x )) = 60 x − 50 + 50 = x 60 60 and G(F (x )) = x + 50 − 50 = x. 60 7. 2.5 acres; Sample explanation: I evaluated G(c) for c = 200. 8. a. f −1 (x ) = x + 10 2 b. g −1(x) = 4x − 5 c. h −1(x) = 6x + 8 d. j−1 (x ) = −x + 2 5 9. D 10. F(50) = 50 = 6 1 ; The distance on 8 4 a map between 2 points that are actually 50 miles apart is 6 1 inches. 4 11. a. G(m) = 8m; Sample explanation: I solved the equation m = a for a. 8 Then I replaced a with G(m). b. The inverse function represents the actual distance in miles between 2 points that are m inches apart on the map. 12. G(3) = 8(3) = 24; The actual distance between 2 points that are 3 inches apart on the map is 24 miles. 13. 36 miles; Sample explanation: I evaluated G(m) for m = 4 1 . 2 14. C 15. a. f ( g ( x )) = 5 x + 3 − 3 = x + 12 5 They are not inverses. b. f ( g ( x )) = 2 x − 6 + 3 = x 2 x g ( f ( x )) = 2 + 3 − 6 = x 2 They are inverses. c. f ( g ( x )) = 2 1 x + 4 − 4 = x 2 1 g ( f ( x )) = [2( x − 4)] + 4 = x 2 They are inverses. d. f(g(x)) = −x − 3 + 3 = −x They are not inverses.

(

)

(

)

(

)

(

)


97



ACTIVITY 6 Continued continued

y

f(x) inverse of f(x)

2 4

2

–2

x

6

–2 –4

b. Sample explanation: List several ordered pairs from f(x), including the vertex. Switch the coordinates of each ordered pair to get new ordered pairs for the inverse function. Graph and connect the ordered pairs. c. f(x): domain: {x | x ∈ }, range: {y | y ∈ , y ≥ 2}; inverse of f(x): domain: {x | x ∈ , x ≥ 2}, range: {y | y ∈ } d. No. Sample explanation: For each value of x greater than 2, there is more than 1 output for each input. y 20.

x f(x)

−2 4

0

1 2

−1

3 0

19. a. Graph the absolute value function f(x) = | x | + 2. b. Graph the inverse of f(x) on the same coordinate plane. Explain how you graphed the inverse. c. Give the domain and range of f(x) and its inverse using set notation. d. Is the inverse of f(x) a function? Explain your answer.

2

4

x

4

22. The graph of a function es through the point (−3, 4). Based on this information, which point must lie on the graph of the function’s inverse? A. (−4, 3) B. (−3, 4) C. (3, −4) D. (4, −3) 23. Explain why the functions f(x) and g(x), graphed below, are not inverse functions.

–4

2

–2

4

x

–2

y f(x)

g(x)

4

f(x)

2 –4

2

–2

4

–4

–2 –4

24. A function h(x) has two different x-intercepts. Is the inverse of h(x) a function? Explain your answer.

Construct Viable Arguments and Critique the Reasoning of Others 26. A student says that the functions f(x) = 2x + 2 and g(x) = 2x − 2 are inverse functions because their graphs are parallel. Is the student’s reasoning correct? Justify your answer.

4 f -1(x)

x

4

–4

–4

y

b. 4

98

2 –4

–2

–2

4

x

2 –4 g(x) = g-1(x)


98

x


2

–2 –4

y

25. Describe a method for determining whether a function f(x) is its own inverse.

inverse

–2

21. Graph each function and its inverse on the same coordinate plane. a. f(x) = 2x + 4 b. g(x) = −x − 2

18. Find the inverse of each function. Then use the definition of inverse functions to that the two functions are inverses. a. f(x) = −3x + 3 b. g(x) = 0.25x + 0.6

y

2

21. a.

5 5

20. Graph the inverse of the function shown below.

4

–4

16. Use a graph to determine which two of the three functions listed below are inverses. a. f (x ) = 2 x + 6 b. g (x ) = 3 x − 6 2 3 c. h(x ) = 3 x − 9 2 17. Write the inverse of the function defined by the table shown below.


22. D 23. Sample answer: The graphs of the functions are not symmetric about the line y = x. 24. No. Sample explanation: The graph of h(x) es through the points (c, 0) and (d, 0), and the graph of the inverse of h(x) must through the points (0, c) and (0, d). For the inverse of h(x), there are 2 outputs, c and d, for the input 0. 25. Sample answer: (1) Let y = f(x), (2) interchange the x and y variables, (3) solve for y, and (4) let y = f −1(x). If the rules for f(x) and f −1(x) are the same, then the function is its own inverse.

26. No. Sample justification: The graphs are parallel because both have a slope of 2. However, f(x) and g(x) are not inverse functions because f(g(x)) = 2(2x − 2) + 2 = 4x − 2 and g(f(x)) = 2(2x + 2) − 2 = 4x + 2.



4


6

–4

Inverse Functions Old from New

ACTIVITY 6

19. a–b.


Piecewise-Defined, Composite, and Inverse Functions

Embedded Assessment 2 Use after Activity 6

CURRENCY CONVERSION Kathryn and Gaby are enrolled in a university program to study abroad in Spain and then in South Africa. They realize that they will have to convert US dollars (USD) to euros (EUR) in Spain, and then convert EUR into South African rand (ZAR) for their time in South Africa. They identified a currency exchange service in Spain that will convert D dollars to euros with the function E(D) = 0.64D − 5 and a currency exchange service in South Africa that will convert E euros to rand using R(E) = 12.1E − 10.

2. If Kathryn converts 450 USD in Spain to EUR, then converts that amount in EUR to ZAR, how much will she have in South African rand? Explain the process you used to arrive at your answer. 3. Explain how to compose the functions E and R to answer Item 2. Write the composite function and identify the domain and range. 4. After converting USD to EUR in Spain, Gaby had 139 EUR. Use an inverse function to find how much USD she converted.


14.00 28.00 35.00

b. 0 < M ≤ 1000, (0, 1000], {M|M ∈ , 0 < M ≤ 1000} c. {C|C ∈ , C = 14 or C = 28 or C = 35} d. Shipping Costs to U.S. C

a. Write a piecewise-defined function that gives the cost C in South African rand for shipping a package with a mass of M grams. b. Write the domain of the function using an inequality, interval notation, and set notation. c. Write the range of the function using set notation. d. Graph the function. e. The package containing Kathryn’s bowl has a mass of 283 grams. If she needs to convert euros to rand to pay for the shipping, how many euros will she need? Explain how you determined your answer.

42

Cost to Ship (ZAR)


5. Kathryn buys a wooden bowl as a souvenir while in South Africa. She wants to ship it back to the United States. The table below shows the costs for one shipping service. Shipping Costs to U.S. Cost to Ship (ZAR)

• Piecewise-defined functions • Composition of functions • Inverse functions 1. Domain for E(D) is USD (dollars), and the range is EUR (euros). Domain for R(E) is EUR (euros), and the range is ZAR (rand). 2. 3414.30 ZAR; Sample explanation: E(450) = 0.64(450) − 5 = 283 R(283) = 12.1(283) − 10 = 3414.30 3. R(E(D)) = 12.1(0.64D − 5) − 10 = 7.744D − 70.5, and R(E(450)) = 7.744(450) − 70.5 = 3414.30; domain: USD, range: ZAR 4. D(E ) = E + 5 is the inverse function, 0.64 and D(139) = 225, so Gaby had 225 USD converted. 14 if 0 < M ≤ 100  5. a. C(M) = 28 if 100 < M ≤ 200  35 if 200 < M ≤ 1000

1. For each function, give the units for the domain and range.

Mass of Package (g)

Assessment Focus

Answer Key

Use the information above to solve the following problems. Show your work.

No more than 100 More than 100 and no more than 200 More than 200 and no more than 1000


6. Kathryn and Gaby are shopping for plane tickets back to their home city of Chicago. The average cost of a plane ticket from Johannesburg, South Africa, to Chicago is $1300. The function g(x) = |x − 1300| gives the variation of a ticket costing x dollars from the average ticket price. a. Graph g(x). b. Describe the graph of g(x) as a transformation of the graph of f(x) = |x|. c. At one travel website, all of the ticket prices are within $200 of the average price. Explain how you can use the graph of g(x) to find the least and greatest ticket prices offered at the website.

35 28 21 14 7 200 400 600 800 1000

M

Mass of Package (g)


99

Common Core State Standards for Embedded Assessment 2

e. 3.72 euros; Sample explanation: If the mass of the package is 283 grams, the shipping will cost 35 rand. The inverse of the function that converts euros to rand is E(R) = R +10 . The function E(R) 12.1 converts rand to euros, and E(35) = 35 + 10 ≈ 3.72. 12.1

HSF-BF.A.1 Write a function that describes a relationship between two quantities. HSF-BF.A.1b Combine standard function types using arithmetic operations. HSF-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. HSF-BF.B.4 Find inverse functions. HSF-BF.B.4a Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. HSF-IF.C.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. HSF-IF.C.7b Graph piecewise-defined functions, including step functions and absolute value functions.


99


Embedded Assessment 2 6. a.

Use after Activity 6

CURRENCY CONVERSION

300

200

Mathematics Knowledge and Thinking

150 100

Exemplary

• Clear and accurate

identification of and understanding of function concepts including domain, range, composition, inverse, and function transformations

(Items 1, 3, 4, 5b, 5c, 6b)

50 800 1000 1200 1400 1600 1800

x

Price of Ticket ($)

b. a translation of the graph of f(x) by 1300 units to the right c. Sample explanation: Find the points where the graph of the function intersects the line y = 200. The intersection points are (1100, 200) and (1500, 200), so the least ticket price offered at the website is $1100 and the greatest ticket price is $1500.

TEACHER to TEACHER You may wish to read through the scoring guide with students and discuss the differences in the expectations at each level. Check that students understand the used.

Problem Solving

(Items 3, 4, 5a-d, 6a)

Reasoning and Communication

Incomplete

and accurate identification of function concepts including domain, range, composition, inverse, and function transformations

and partially accurate identification of function concepts including domain, range, composition, inverse, and function transformations


and inaccurate identification of function concepts including domain, range, composition, inverse, and function transformations

• No clear strategy when

unnecessary steps but results in a correct answer

some incorrect answers

• Fluency in creating

• Little difficulty in creating

• Partial understanding of


•

•

•

•

piecewise-defined, inverse, and composite functions to model real-world scenarios Clear and accurate understanding of how to graph piecewise-defined functions and represent intervals using inequalities, interval notation, and set notation

• Precise use of appropriate

(Items 2, 3, 5e, 6b, 6c)

•


• A functional understanding • Partial understanding

strategy that results in a correct answer

math and language to describe function transformation and function composition Clear and accurate explanation of the steps to solve a problem based on a real-world scenario

100 SpringBoard® Mathematics Algebra 2

100

Emerging

• An appropriate and efficient • A strategy that may include • A strategy that results in

(Items 2, 5e, 6c)

Mathematical Modeling / Representations

Proficient

The solution demonstrates these characteristics:

piecewise-defined, inverse, and composite functions to model real-world scenarios Mostly accurate understanding of how to graph piecewise-defined functions and represent intervals using inequalities, interval notation, and set notation

• Adequate description of •

function transformation and function composition Adequate explanation of the steps to solve a problem based on a real-world scenario

how to create piecewisedefined, inverse, and composite functions to model real-world scenarios Partial understanding of how to graph piecewisedefined functions and represent intervals using inequalities, interval notation, and set notation

solving problems

of how to create piecewisedefined, inverse, and composite functions to model real-world scenarios Inaccurate or incomplete understanding of how to graph piecewise-defined functions and represent intervals using inequalities, interval notation, and set notation

• Misleading or confusing

• Incomplete or inaccurate

•

•

description of function transformation and function composition Misleading or confusing explanation of the steps to solve a problem based on a real-world scenario

description of function transformation and function composition Incomplete or inadequate explanation of the steps to solve a problem based on a real-world scenario


Scoring Guide

250


Variation from Average Price ($)

y

Piecewise-Defined, Composite, and Inverse Functions


Variation from Average Ticket Price

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