Practice Lesson 7 Compare Functions - Drauden Point...

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Practice Lesson 7 Compare Functions

Unit 2

Practice and Prob

lem Solvin

gU

nit 2 Functions

Key

B Basic M Medium C Challenge

©Curriculum Associates, LLC Copying is not permitted. 73Lesson 7 Compare Functions

Name: Compare Functions

Lesson 7

Prerequisite: Identify Functions

Study the example showing a function. Then solve problems 1–6.

1 Describe the relationship between the input and output values in the example

2 Can you represent the function in the example with an equation? If so, what equation can you write? If not, why not?

3 In the example function, could one side length ever produce two diff erent perimeters? Explain

Example

The table and graph show the relationship between the length of the sides of a square, in feet, and the perimeter of the square in feet

Side Length (input) 1 2 3 4 5

Perimeter (output) 4 8 12 16 20

The relationship is a function because there is only one output value for each input value

Vocabularyfunction a rule that

assigns exactly one

output to each input

input the number put

into a function

output the number

that results from

applying the function

to the input

Peri

met

er (o

utpu

t)

Perimeters of Squares

Side Length (input)2 4 6 8 91 3 5 7O

x

y

4

2

8

12

14

16

18

20

6

10

7373

B

Possible answer: Each output value, which represents the perimeter, is 4 times the input

value, which represents the side length of the square.

B

Yes; Possible explanation: The function can be

represented by y 5 4x, where y 5 perimeter and

x 5 side length.

M

No. Each side length produces only one perimeter.

It’s not possible to have two different perimeters

for a given side length.

©Curriculum Associates, LLC Copying is not permitted.74 Lesson 7 Compare Functions

Solve.

4 Do the data in this table show a function? If you switch the input and the output values, would the data show a function? Explain

Input 1 2 2 3 3

Output 6 9 11 12 14

5 Substitute values into the equation to complete the table Then state whether the equation represents a function Explain your reasoning

y 5 5x 1 1

x (input) 22 21 0 1 2

y (output)

6 A teacher wrote these numbers on the board: 25, 23, 21, 1, 1, 2, 3, 4, 4, 6 The input–output diagram has been started using the teacher’s numbers to form ordered pairs of a function

Part A: Put the remaining numbers in the ovals to complete the diagram

Part B: If the input and output values were reversed, would the diagram still represent a function? Explain

25

21

23

1

Input Output

74

M

M

29 24 1 6 11

Possible answer: It is a function. Each output is 1 more than 5 times the corresponding

input. So each input has only one output.

C

Possible answer shown in diagram.

Yes; Possible explanation: If the values were reversed, each input would have only one output.

So it would be a function.

2

1

4

4

3

6

Possible answer: The data in the table do not show a function. There is more than one

output for some of the inputs. If you switch the input and output values, each input will

have exactly one output, so it would be a function.

28©

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Unit 2 Fun

ctions

Unit 2



Name: Lesson 7

1 What do the rates of change in the example represent?

2 What does it mean in the context of the example that Alyssa’s rate of change is greater than Sarah’s?

3 Write ordered pairs for the initial values of each function in the example Tell what the initial values represent

Example

Compare the rates of change for these two functions Which function has a greater rate of change?

Tota

l Sav

ings

($)

Sarah’s Savings

Weeks2 4 6 8 91 3 5 7O

x

y

8

4

16

24

28

32

36

40

12

20

Tota

l Sav

ings

($)

Alyssa’s Savings

Weeks2 4 6 8 91 3 5 7O

x

y

8

4

16

24

28

32

36

40

12

20

Alyssa’s rate of change is greater than Sarah’s

vertical change ·············· horizontal change 5 8 ·· 1 5 8 vertical change ·············· horizontal change 5 6 ·· 1 5 6

Vocabularyrate of change the rate

at which one quantity

increases or decreases

with respect to a change

in the other quantity It is

the ratio of the vertical

change to the horizontal

change on a graph

initial value the

starting value of a

function

Interpret and Compare Rates of Change

Study the example problem showing how to compare rates of change. Then solve problems 1–5.

75

BPossible answer: The rates of change represent the

amount of money each person saves per week.

B

Alyssa saves more per week than Sarah.

M

Sarah’s savings: (0, 8), Alyssa’s savings: (0, 0); Possible

answer: Sarah had $8 before starting to save, while

Alyssa did not have any money to start with.


Solve.

4 The table shows the weight gain of a kitten over a 5-week period The graph shows the weight gain of a second kitten over the same period Compare the rates of change for these two functions

Kitten AWeek Weight (oz)

0 3

1 7

2 11

3 15

4 19

5 23

5 Sonya sells bracelets once a month at a fl ea market The table shows her profi ts for a 5-month period

SonyaMonth 1 2 3 4 5

Total Profit ($) 30 60 90 120 150

a. Kirsten sells bracelets once a month at a different flea market The rate of change for her profits is $10 per month Complete the table and the graph to show her total profits

Kirsten

Month 1 2 3 4 5

Total Profit ($) 10

b. Sonya says that her profit is increasing 4 times as fast as Kirsten’s profit Do you agree? Explain

Wei

ght (

oz)

Kitten B

Weeks2 4 6 8 91 3 5 7O

x

y

4

2

8

12

14

16

18

6

10

Tota

l Pro

�t ($

)

Kirsten

Month2 4 6 8 91 3 5 7O

x

y

10

5

20

30

35

40

45

50

15

25

76

M

The rate of change for Kitten A is 4 oz per week, and the rate of change for Kitten B is

3 oz per week. Kitten A is gaining weight faster than Kitten B.

C

No; Possible explanation: Sonya’s rate of change is $30 per month, and Kirsten’s rate

of change is $10 per month. Sonya’s profit is increasing 3 times as fast as Kirsten’s.

20 30 40 50

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Solving

Unit 2 Fun

ctions Unit 2



Name: Lesson 7

1 What do the initial values mean in the context of the example problem?

2 Do the functions in the example show positive or negative rates of change? Explain

3 Write an equation for each function, where x is the number of months and y is the amount owed

Mr Allen’s plan:

Mr Jessup’s plan:

Example

Mr Allen bought a new computer His monthly payment plan is shown in the table

Month 0 1 2 3 4 5 6 7

Amount Mr. Allen Owes ($)

560 480 400 320 240 160 80 0

Mr Jessup buys a new computer for $400 He makes monthly payments of $40 until the computer is paid for Compare the initial values and rates of change of each function

You can graph both functions to show that the amount Mr Allen owes starts at $560 and decreases $80 per month The amount that Mr Jessup owes starts at $400 and decreases $40 each month

Mr Allen’s initial value is $160 more than Mr Jessup’s Mr Allen’s rate of change is greater than Mr Jessup’s rate of change

Am

ount

Ow

ed ($

)

Payment Plans

Months2 4 6 8 9 101 3 5

Mr. Allen

7Ox

y

160

80

320

480

560

640

720

240

400

Mr. Jessup

Compare Negative and Positive Rates of Change

Study the example problem showing how to compare two functions. Then solve problems 1–6.

77

B

The initial values are the initial costs of the computers.

negative; As the numbers on the x-axis increase, the numbers on the y-axis decrease.

B

M

y 5 560 2 80x

y 5 400 2 40x


Solve.

4 Below are two companies’ rates to rent a bicycle How much does it cost per hour to rent a bicycle at Company A? What is the cost to rent a bicycle for 6 hours from each company?

Company A: c 5 5h 1 4, where c 5 total cost (in dollars) and h 5 number of hours

Company B: $6 per hour per bicycle

5 Roy wants to buy a new television for $300 Two stores off er diff erent payment options Compare the initial values and rates of change

Show your work.

Solution:

6 Most plumbing companies charge a fee to come to your house plus a charge per hour of work The fees and charges for two plumbing companies are shown

Write an equation for each company, where c 5 total cost (in dollars) and h 5 number of hours Explain what the initial values and rates of change mean in this context

Company A:

Company B:

Store A Payment PlanMonth 0 1 2 3 4 5 6

Amount Owed ($) 300 250 200 150 100 50 0

Store B Payment PlanPay $100 at the time of purchase Pay $50 per month until the television is paid for

Company AFee: $50Charge per hour: $40

Company BFee: $25Charge per hour: $50

78

It costs $5 per hour to rent a bicycle at Company A; At Company A when h 5 6,

c 5 5(6) 1 4 5 34, so it costs $34 to rent a bike for 6 hours. At Company B, 6(6) 5 36,

so it costs $36 to rent a bike for 6 hours.

M

C

Possible work: Initial values: Store A is $300 in the table. Store B is $200 ($300 2 $100). Rates of change: Store A is $50 and Store B is also $50.

Store A’s initial value is $100 greater. The rates of change are the same.

The initial value is the fee, and the charge per hour is the rate of change.

c 5 50 1 40h

c 5 25 1 50h

MM

30©

Cu

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m A

ssociates, LL

C

Copyin

g is not perm

itted.Practice an

d Problem

Solving

Unit 2 Fun

ctions

Unit 2



Name: Lesson 7

1 A hardware store charges a $30 rental fee and $15 per day to rent a power washer Which equation correctly relates the total cost y to rent the washer for x days?

A y 5 15 1 30x C y 5 30 2 x ·· 15

B y 5 30 1 15x D y 5 15 2 x ·· 30

Compare Functions

Solve the problems.

3 Alma borrows money from her mom to buy a $150 bike She gives her mom $40 at the time of purchase and continues to pay her $10 each month until the bike is paid for in full Alma wrote this equation to represent the amount y that she will have paid her mom after x months

Equation: y 5 40x 1 10

Is her equation correct? How did she get that equation? If it is not correct, write a correct equation

2 Tony drives 18 miles to pick up his friend at his house Then he drives at a constant speed of 40 miles per hour to a state park to go hiking Let y represent the number of miles that Tony drives after x hours Which of the following statements are true? Select all that apply

A The relationship can be represented by the equation y 5 40x 1 18

B If Tony travels for 1 5 hours, he will have driven a total of 60 miles

C The initial value is 18 miles

D The rate of change is negative

What do the parts of each equation represent?

How do you determine the initial value and rate of change?

How does an equation show a rate of change?

79

B

B

No; Alma thought 40 was the rate of change and not the initial value. The correct

equation is y 5 10x 1 40.

M


Solve.

4 The rates for two airport shuttles are shown below

Quick ShuttleRates for shuttle • $30 for passenger pickup • $0 50 for each mile driven

Part AWhich shuttle service has a greater initial value? Which service has the greater rate of change? Explain what the greater initial value and greater rate of change mean

Show your work.

Solution:

Part BWhich shuttle company would cost less for a 25-mile trip?

Show your work.

Solution:

How does the graph show the initial value?

Cost

($)

We-Drive Shuttle

Miles10 20 30 40 50O

x

y

20

10

40

60

70

30

50

80

Possible work:Quick Shuttle: Initial value 5 $30. Rate of change 5 $0.50 per mile.We-Drive Shuttle: Initial value 5 $20. Rate of change 5 $10 for 10 miles, or $1 per mile.

Quick Shuttle has a greater initial value. This means that the cost to be

picked up is more. We-Drive Shuttle has the greater rate of change. This means

that they charge more per mile than Quick Shuttle does.

We-Drive Shuttle: $20 1 25($1) 5 $45Quick Shuttle: $30 1 25($0.50) 5 $42.50

Quick Shuttle

C

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