Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value

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HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value

I Can find and interpret initial value and rate of change.

Spark Your Learning

Bonnie has decided to join a gym for the next 3 months. Fabulous Fitness charges $45 per month. Greg’s Gym charges $99 at the end of each quarter. How can she compare the two options to determine the better value?
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 1
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 2

Turn and Talk What is the rate of change per month for each membership and how do you know?

Build Understanding

Many real-world functional relationships have only nonnegative inputs and rates of change that are constant. You can use initial value and the rate of change to analyze such functions. Initial value is the value of a function when the input is zero. A constant rate of change describes the slope or steepness of a graph, or how quickly two quantities change.

1. Bonnie continues to research gym options and finds out that Fabulous Fitness charges no sign-up fee, while Greg’s Gym charges $50 for the initial sign-up. She makes the following table to help her compare the two options.
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 3

A. How much would Bonnie need to pay just to join each gym?
____________________________
Answer:
Bonnie needs to pay $0 to just join the fabulous fitness.
Bonnie needs to pay $50 to just join Greg’s gym.

B. How much would Bonnie need to pay per month at each gym?
____________________________
____________________________
Answer:
Bonnie needs to pay per month to the fabulous fitness is $45.
Bonnie needs to pay per month to Greg’s gym is $0
Greg’s gym has no monthly fee it has only the sign-up fee and the third-month fee,
The third month’s fee is $99

C. Use an equation to calculate the total cost for three months of each gym option. Show your work.
____________________________
____________________________

D. In the equation y = mx + b, the slope m represents the constant rate of change, or cost per month. What does the y-intercept b represent on the graph of the function?
____________________________

Turn and Talk Why is it important to consider both initial value and rate of change when modeling functions in the real world?

Step It Out

2. The owner of Fabulous Fitness wants to buy name tags for her staff. She writes the following linear function to represent the cost y for the quantity of name tags x from an online vendor.
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 4
y = $2.00x + $7.00
A. What is the rate of change?
____________________________
Answer:
Given that the equation is
y = $2.00x + $7.00
Rate of change = slope
The slope of the equation is $2.00
Rate of change = $2.00

B. What is the initial value and what could it represent?
____________________________
____________________________
Answer:
Given that the equation is
y = $2.00x + $7.00
The given equation is in the form of y = mx + b
Initial value = b
b = $7.00
Initial value = $7.00

C. The function representing the cost of the name tags purchased is shown in the graph. Complete the coordinates of the point shown on the graph.
____________________________
(10, HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 5)
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 6
Answer:
From the graph, the coordinate points are (10, 26)

D. Explain what the point means.
It will cost ___ for ___ name tags.
Answer:
It will cost $26 for 10 name tags.

E. Substitute 25 for x in the equation and solve for y.
y = 2x + 7
y = 2(HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 5) + 7
y = HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 5
Answer:
Given that the equation is
y = 2x + 7
Substitute x = 25
y = 2(25) + 7
y = 50 + 7
y = 57
F. What point represents the values in Part E?
(HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 5, HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 5)
Answer:
The values represented in part E are (25, 57).
G. What is the meaning of the point?
It will cost ___ for ____ name tags.
Answer: It will cost $57 for 25 name tags.
H. How many name tags can the owner purchase for $93.00?
_________________
Answer:
Given that the equation is
y = 2x + 7
Substitute y = 93
93 = 2x + 7
93 – 7 = 2x
86 = 2x
x = 86/2
x = 43
For 43 name tags can the owner purchase for $93.00.
I. How did you determine the answer to Part H?
_________________
Answer: We determine the answer by substituting the y value = 93 in the given equation then we get the number of tags.
J. Is this function discrete or continuous? Why?
_________________
_________________
Answer: The function continues. Because the number of tags and the cost of tags are continuously increasing.

3. The following table shows how the value of a new car depreciates, or decreases, over time.
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 7

A. What is the purchase price of the car?
___________________
B. Calculate the slope using the points
(1, 20,000) and (3, 17,000).
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 8
m = HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 9
C. Calculate the slope using the points
(3, 17,000) and (4, 15,000).
m = HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 9
D. Compare the two slopes. What do you notice?
They (are / are not) equal.
E. Explain whether or not the depreciation of the car is a linear function.
The change (is /is not) constant, so the depreciation (is / is not) a linear function.
Answer:
A. The purchase price of the car is 24,000.
B. Given that the points are
(x1, y1) = (1, 20,000)
(x2, y2) = (3, 17,000)
Slope = m = y2 – y1/ x2 – x1
= 17,000 – 20,000/3-1
= -3000/3
= -1500
m = -1500
C. Given that the points are
(x1, y1) = (3, 17,000)
(x2, y2) = (4, 15,000)
Slope = m = y2 – y1/ x2 – x1
= 15,000 – 17000/4 – 3
= -2000/1
= -2000
m = -2000.
D. First slope = -1500
Second slope = -2000
The two slopes are not equal.
E. The rate of change is not equal, so the depreciation is not a linear function.

Check Understanding

Question 1.
Sam has $37 to spend at a festival The price of admission is $10 and tickets to each attraction cost $0.75. How many tickets can he buy?
A. What is the initial value, and what does it represent?
Answer:
Given that,
Sam has $37 to spend at a festival
The price of the admission = $10.
The ticket coat = $0.75.
The cost of one ticket = $10 + $0.75 = $10.75
From the given information the equation is y = 10.75x + $37.
The initial value = $37
It represents the Sam spent at the festival.

B. What is the rate of change?
Answer:
The rate of change = slope
Slope = $10.75
The rate of change = $10.75
It represents the cost of 1 ticket and the admission.

C. What is the maximum number of tickets Sam can purchase? Explain.
Answer:
Given that,
Sam has $37 to spend at a festival
The price of the admission = $10.
The ticket coat = $0.75.
The cost of one ticket = $10 + $0.75 = $10.75
Therefore $37/10.75 = 3.441
Sam can purchase a maximum of 3 tickets.

D. Complete the table for Sam’s trip to the festival.
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 10
Answer:
y = 10.75x
If x = 1 then y = 10.75.
If x = 5 then y = 10.75(5) = 53.75
If x = 10 then y = 10.75(10) = 107.50.
If x = 15 then y = 10.75(15) = 161.25
HMH-Into-Math-Grade-8-Module-6-Lesson-3-Answer-Key-Interpret-Rate-of-Change-and-Initial-Value-10

On Your Own

Reason Use the following information to answer Problems 2-3.

Jeffrey is shopping for a new mobile device. He needs a device as well as a data plan. He is deciding between the two options:
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 11

HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 12

Question 2.
Which option has the lower initial value? Explain.
Answer:

Question 3.
Which option has the lower constant rate of change? Explain.
Answer:

Question 4.
Construct Arguments Toni is deciding whether to subscribe to app A or app B for streaming music. App A offers listeners a $4 startup fee plus $0.15 per song. App B offers listeners no startup fee and charges $0.40 per song.
A. Which function has a greater initial value? Explain.
Answer:

B. Which app has a greater rate of change? How can you tell?
Answer:

C. Toni wants to purchase 20 songs per month. Which app should she choose? Why?
Answer:
The charges for songs in app A = $4 + $0.15 = $4.15
For 20 songs in app A = 20 x $4.15 = $83
The charges for songs in app B = $0 + $0.40 = $0.40
For 20 songs in app B = 8
Therefore she can choose app B because the cost for the 20 songs in app A is less than the cost of 20 songs in app B.

The graph shows the value of a comic book over time after a collector buys it. Use the graph for Problems 5—8.

Question 5.
Is the rate of change positive or negative? What does this mean?
Answer:
For the positive rate of change, the measuring graph increases.
For the negative rate of change, the measuring graph decreases.
The rate of change is positive. Because the time and the value are increasing.

HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 13

Question 6.
What does the point (15, 40) mean in the context of this linear function?
After HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 14 years, the comic book is worth HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 14 dollars.
Answer: After 15 years, the comic book is worth 40 dollars.

Question 7.
A. Where do you find the initial value on the graph shown?
_____________________
Answer:

B. What is the initial value? What does it represent?
_____________________
Answer:

C. What is the slope of the line? What does it represent?
_____________________
Answer:

D. Model with Mathematics Write the equation of the line.
_____________________
Answer:

E. Use the linear equation to find the value of the comic book after 25 years.
_____________________
Answer:

Question 8.
A. If you extend the line, what is its x-intercept, the x-coordinate where the line crosses the x-axis? _________________
Answer:

B. Does the x-intercept make sense in the context of the problem? Explain.
_________________
Answer:

I’m in a Learning Mindset!

What challenge did I have to overcome to interpret the value of the comic book over time using the graph?

Lesson 6.3 More Practice/Homework

Use the information to answer Problems 1—2.

The following table shows the cost to order muffins to be delivered from a bakery.
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 15

HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 16

Question 1.
Use Structure What is the rate of change, and what does it represent?
Answer:
From the given table
(x1, y1) = (1,17)
(x2, y2) = (5, 25)
Rate of change = slope = y2 – y1/x2 – x1
= 25 – 17/5 – 1
= 8/4
= 2
Rate of change = 2
It represents the cost of the order.
For 1 muffin = $17
Delivery fee = $15
Therefore $17 – $15 = $2

Question 2.
What is the initial value, and what does it represent?
Answer:
The equation is y = mx + b
x = 1 then y = 17
m = 2
17 = 2(1) + b
17 = 2 + b
17 – 2 = b
b = 15
Initial value = $15
It represents the delivery fee.

Use the information to answer Problems 3—5.

The following linear function represents the cost of a home printer plus ink cartridges where x represents the number of ink cartridges purchased and y represents the total cost.
y = 25x + 115

Question 3.
What is the rate of change, and what does it represent?
Answer:
Given that the equation is y = 25x + 115.
It is in the form of y = mx + b
Rate of change = slope
The slope of the equation = 25.
It represents the cost of ink Cartridges

Question 4.
How much does the printer itself cost?
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 17
Answer:
Given that the equation is y = 25x + 115.
It is in the form of y = mx + b
The cost of printer itself is 115

Question 5.
How much will the purchaser spend buying the printer and 6 ink cartridges?
Answer:
Given that the equation is y = 25x + 115.
It is in the form of y = mx + b
Where x = number of ink cartridges = 6
Substitute 6 in the equation.
y = 25(6) + 115
y = 150 + 115
y = 265
The cost of the printer and 6 ink cartridges is $265.

Question 6.
Open Ended Write a scenario that can be modeled by the linear function y = 25x + 40.
Answer:
Given that the linear equation is y = 25x + 40
The equation is in the form of y = mx + b
Where, m = slope = 25.
b = initial value = 40.

Test Prep

Question 7.
Jose starts a savings account with $80. At the end of 6 months, the account has $350. Assuming there is a constant rate, how much did Jose save per month?
A. $270
B. $72
C. $60
D. $45
Answer:
Given that,
Jose starts a savings account with $80.
At the end of 6 months, the account is $350.
After 6 months Jose saves is $350 – $80 = $270
Option A is the correct answer.

Question 8.
Camille wants to set up a hot chocolate stand to raise money. She spends $40 on hot chocolate and $0.25 for each paper cup. She uses the equation y = 0.25x + 40 to keep track of her spending. If Camille has $60 in donations to spend, what is the maximum number of paper cups she can buy?
A. 5 cups
B. 80 cups
C. 240 cups
D. 400 cups
Answer:
Given that the equation is
y = 0.25x + 40
Substitute y = $60
$60 = 0.25x + 40
$60 – 40 = 0.25x
$20 = 0.25x
x = 20/0.25
x = 80 cups.
The maximum number of paper cups that she can buy is 80 cups.
Option B is the correct answer.

Question 9.
Admission to a carnival costs $15. Each game costs $2 to play. Which statements about the function modeling the cost to attend the carnival are true? Select all that apply.
A. The slope is positive.
B. The initial value is $15.
C. The rate of change is $15.
D. The graph has a y-intercept of 2.
E. The graph passes through the origin.
Answer:

Spiral Review

Question 10.
Solve 4x – 12 = -7x – 111.
Answer:
4x – 12 = -7x – 111
4x + 7x = 12 -111
11x = -99
x = -99/11
x = -9

Question 11.
Calculate the slope of the line that passes through (3, 2) and (-7, 4).
Answer:
(x1, y1) = (3, 2)
(x2, y2) = (-7, 4)
slope = (y2 – y1)/(x2 – x1)
m = (4 – 2)/(-7-3)
m = 2/-10 = -1/5
Thus the slope is -1/5

Question 12.
The table shows a proportional relationship. Complete the table.
HMH Into Math Grade 8 Module 6 Lesson 3 Answer Key Interpret Rate of Change and Initial Value 18
Answer:
The unit rate is 1: 4
y = 4x
y = 4(2) = 8
HMH-Into-Math-Grade-8-Module-6-Lesson-3-Answer-Key-Interpret-Rate-of-Change-and-Initial-Value-18

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