Into Math Grade 8 Module 6 Lesson 5 Answer Key Compare Functions

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HMH Into Math Grade 8 Module 6 Lesson 5 Answer Key Compare Functions

I Can compare functions presented in equations, tables, graphs, and verbal descriptions.

Step It Out

1. Four friends want to buy a drone for a science fair project. To afford the drone, they decide to buy a used lawn mower and start a lawn-care business. Each of the friends represents the profit of their business differently.

Describe their lawn-care business, including the meaning of the slope and y-intercept of the function modeling their business.
The unit rate is _____.
The initial cost of the lawn mower is _____.
The friends’ profit is positive after they have mowed _____.

Turn and Talk Which representation do you prefer to interpret? Explain your preference. Which representation best illustrates the friends’ process and their goal? Which contains the most information?

Choosing how to represent a linear function depends upon the situation and what you need to know. When linear functions are represented differently, finding the slope and y-intercept for each can help compare information.

2. The four friends researched companies to host a website with the results of their science fair project. They need to decide which company to use. Some of the companies charge a one-time fee for a domain name.

A. What are the slope and y-intercept for each representation?
Company A: slope = y-intercept =
Company B: slope = y-intercept =
Company C: slope = y-intercept =
Company D: slope = y-intercept =

B. What does the slope represent for each company?
_____________________
C. What does the y-intercept represent for each company?
_____________________

D. Peter says that Company C will always be the least expensive option because the domain name is free. Is Peter correct? Peter (is /is not) correct. Company ___ is the least expensive option for the first
___ months, but then starts to become more expensive than the other plans because the ____ is greater.

E. Marley says that Company A charges $2 per month for their Web page. Is Marley correct? Marley (is /is not) correct. In the table, the cost increases by ___ for each additional month. This means the rate
of change (is / is not) constant. The slope of the line is ___.

Turn and Talk Which representation do you prefer to use to decide which company offers the best plan long term? Explain your choice.

Check Understanding

Use the given information to answer Problems 1-3.

Lab Thinkers of Tomorrow has this table displayed on its website that shows its costs.

Question 1.
Peter wants to join a science club that sends him a new box of science experiments each month. Which club has the higher sign-up fee? Explain.
Answer:

Question 2.
What is the slope of the graph of each function? What does the slope represent?
Answer:

Question 3.
If Peter wanted to join a club for only six months, which club would be the better deal? Why?
Answer:

On Your Own

Use the given information to answer Problems 4-6.

Four aerial photographers will need 6 batteries to operate their drones. Each found a different website that sells rechargeable batteries.

Question 4.
Attend to Precision For each website, find the cost of 6 rechargeable batteries and a charger. Which website has the best price?
Website A charges ____ for 6 batteries and a charger.
Website B charges ____ for 6 batteries and a charger.
Website C charges ____ for 6 batteries and a charger.
Website D charges ___ for 6 batteries and a charger.
Website ____ has the best price.
Answer:

Question 5.
The fixed value represents _________
The rate of change represents _________.
Answer:

Question 6.
Determine the cost of the charger from Website A. ______
Determine the cost of the charger from Website C. ______
Determine the cost of the charger from Website D. _____
Answer:

Use the given information to answer Problems 7-11.

Leon and Jacey work for a florist. They each program a drone to deliver boxes of flowers to neighbors. With Leon’s program, he must manually attach the box to the drone. With Jacey’s program, the drone can pick up one box at a time from a pile.

Question 7.
How much time does it take Leon to attach a box?
_____________________
Answer:

Question 8.
How much time does Jacey’s drone take to pick up a box?
_____________________
Answer:

Question 9.
Reason Which program has the drone traveling faster? How do you know?
_____________________
Answer:

Question 10.
What does the y-intercept represent in this situation?
_____________________
Answer:

Question 11.
Open Ended Why might the slower delivery program be a better choice?
_____________________
Answer:

For Problems 12—14, find the slope and y-intercept for each linear function.

Question 12.

The slope is _____.
The y-intercept is ___.
Answer:
The coordinates points from the graph are (0, 5), (1, 7), (2, 11).
x1 = 0, y1 = 5, x2 = 1, y2 = 7
Slope = y2 – y1/x2 – x1
= 7 – 5/1 – 0
= 2/1
= 2
The equation is y = mx + c
m = slope
c = y intercept
y = 2x + c
c = y – 2x
c = 5 – 2(0)
c = 5
The slope = 2
The y intercept = 5.

Question 13.

The slope is _____.
The y-intercept is ___.
Answer:
The equation is y = mx + c
From the table x1 = 1, x2 = 2, y1 = 6, y2 = 7
Slope = y2 – y1/ x2 – x1
= 7 – 6/2 – 1
= 1/1
= 1
y-intercept = c = y – 1x
c = 6 – 1(1)
c = 5
Slope = 1
y-intercept = 5

Question 14.
An online store sells music. Each download costs $0.99, and there is a one-time membership fee of $10.
Answer:
Given that,
Each download costs $0.99.
one-time membership fee of $10
Slope = $0.99
y-intercept = $10.

Question 15.
Reason The table and graph show the hourly charges for two parking garages. Compare the charges by finding slopes and y-intercepts and interpreting those values in the context of the situation.

Answer:
The equation is y = mx + b
In the Garage A x1 = 0, x2= 1, y1 = 5, y2 = 7
Slope = m = y2 – y1/ x2 – x1
= 7 – 5/1- 0
= 2/1
= 2
y = 2x + b
b = y – 2x
b = 5 – 2(0)
b = 5
The equation is y = 2x + 5
The slope of the Garage A is 2 and the y-intercept = 5
Garage B the two coordinate points are (0,7), and (1,8).
x1 = 0, x2= 1, y1 = 7, y2 = 8
Slope = m = y2 – y1/ x2 – x1
= 8 – 7/1 – 0
= 1/1
= 1
y = 1x + b
b = y – 1x
b = 7 – 1(0)
b = 7
The equation is y = 1x + 7
The slope of the Garage B is 1 and the y-intercept = 7

Lesson 6.5 More Practice/Homework

Use the given information to answer Problems 1-5.

To raise money to buy the microscope shown, four friends decide to start a dog-walking business.

Question 1.
Reason Do the table, graph, equation, and words represent the same situation? Explain.
Answer:
The table, graph, equation, and words represent the same situation.
From the table x1 = 8, x2 = 9, y1 = 19, y2 = 22
Slope = m = y2 – y1/ x2 – x1
= 22 – 19/9 – 8
= 3
The equation is y = mx + b
m = 3
y = 3x + b
b = y – 3x
b = 19 – 3(8)
b = -5
Dylan’s equation is y = 3x – 5
From the given graph the Cliffs representation is y = 3x – 5
Andre’s representation is y = 3x – 5
In Tian’s representation, $3 is the slope and $5 is the y-intercept.
The equation is y = 3x + 5

Question 2.
What is the slope in this situation and what does it represent?
Answer: $3 is the slope in this situation. It represents the charges for taking the dog out for walking after school.

Question 3.
What is the y-intercept in this situation and what does it represent?
Answer: – 5 is the y-intercept in this situation. It represents the cost of the treat’s boxes.

Question 4.
How many dogs do the friends need to walk in order to have enough money to buy the microscope? Which representation did you use to answer the question?
Answer:
The cost of the microscope is $254.
The charge for 1 dog = $3
Therefore 254/3 = 84.66
There are 85 dogs do the friends need to walk to buy a microscope.

Question 5.
How many dogs do they need to walk in order to pay for the box of treats?
Answer:
The cost of box of treats = $5
The charge of each dog = $3
They can walk 1 dog in order to pay for the box of treats.

Test Prep

Martha earns $15 per hour for babysitting. The parents always give her a $5 tip. Use the information to answer Problems 6-8.

Question 6.
What is the slope in this ____ situation?
___ dollars per hour
Answer: The slope of the situation is $15 per hour.

Question 7.
What is the y-intercept in this situation?
___ dollars
Answer: The y-intercept of the situation is 5 dollars.

Question 8.
How many hours did Martha babysit if she made $65?
___ hours
Answer:
Martha earns $15 per month on babysitting.
She gets $5 per hour as a tip.
Total she gets = $15 + $5 = $20
For $65 number of hours = 65/20 = 3.25
Therefore when she works for 3 hours and 25 minutes, she gets $65.

Question 9.
Marlene started with a necklace that was 14 inches long. For every bead she adds, the necklace gets \(\frac{1}{4}\) inch longer. What is the slope of the function that represents this situation?
A. \(\frac{1}{4}\)
B. 3.5
C. 4
D. 14
Answer:
From the given data,
The slope is \(\frac {1}{4} \) inch.
Option A is the correct answer.

Question 10.
Two companies rent kayaks for up to 12 hours per day. Company A charges $10 per hour and $7 per day for safety equipment. Company B’s daily charges for x hours of kayaking are represented by the equation y = 7x + 10. Which company has a greater fixed cost for a 12-hour day of kayaking?
A. Company A
B. Company B
C. They have the same fixed cost.
D. cannot be determined
Answer:
Company A rents kayaks for up to 12 hours per day and charges $10 per hour and $7 per day for safety equipment.
Company A equation is $10(12) + 7 = $127.
Given that the equation for the company B is y = 7x + 10
x = number of hours.
After 12 hours y = 7(12) + 10 = $94
Company A has the greatest fixed price.
Option A is the correct answer.

Question 11.
Which has the greater rate of change, the function in the table or the function y = 3x + 2?

A. the function in the table
B. the function y = 3x + 2
C. The rate of change is the same.
D. cannot be determined
Answer:
Rate of change of a table = slope = y2 – y1 /x2 – x1 = 4 – 2/1 – 0
= 2/1
= 2
slope = 2
From the equation the slope is 3
The function y = 3x + 2 has a greater rate of change.
Option B is the correct answer.

Spiral Review

Question 12.
What is the solution to the equation 2w – 3 = 11?
Answer:
Given that the equation is
2w – 3 = 11
2w = 11 + 3
2w = 14
w = 14/2
w = 7.

Question 13.
Is the graph of the number of baskets a player shoots over time during a game discrete or continuous?
Answer:
We need to find the type of graph that the number of baskets a player shoots overtime during a game.
The number of shoots must be an integer that the number cannot be a decimal.
So, the graph is a discrete graph.

Question 14.
Brett spent $25 for a tennis racket and pays $5 each time she plays a game. What is the initial value? What is the slope?
Answer:
Given that,
Brett spent $25 on a tennis racket.
She plays a $5 for each time.
From the data the equation is
y = 5x + 25
The above equation is in the form of y = mx + c
Where,
m = slope = 5
c = y intercept = initial value = 25
The initial value is $25
Slope = $5