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Go Math Grade 8 Chapter 6 Functions Answer Key
Test and improve your knowledge by using Go Math Grade 8 Chapter 6 Functions Solution Key. The Go Math Grade 8 Chapter 6 Functions Answer Key consists of the topics like Identifying and representing functions, describing functions, analyzing graphs, etc. Use HMH Go Math Grade 8 Answer Key for the best practice of maths. After completion of your preparation test yourself by solving the problems given in the model quiz.
Lesson 1: Identifying and Representing Functions
- Identifying and Representing Functions – Page No. 158
- Identifying and Representing Functions – Page No. 159
- Identifying and Representing Functions – Page No. 160
Lesson 2: Describing Functions
- Describing Functions – Page No. 164
- Describing Functions – Page No. 165
- Describing Functions – Page No. 166
Lesson 3: Comparing Functions
- Comparing Functions – Page No. 170
- Comparing Functions – Page No. 171
- Comparing Functions – Page No. 172
Lesson 4: Analyzing Graphs
Model Quiz
Mixed Review
Guided Practice – Identifying and Representing Functions – Page No. 158
Complete each table. In the row with x as the input, write a rule as an algebraic expression for the output. Then complete the last row of the table using the rule.
Question 1.
Type below:
_______________
Answer:
Explanation:
Unit Cost of ticket = 40/2 = 20
Total cost = 20x where x is the number of tickets.
x = 20x
10 = 20(100) = 200
Question 2.
Type below:
_______________
Answer:
Question 3.
Type below:
_______________
Answer:
Determine whether each relationship is a function.
Question 4.
_______________
Answer:
Function
Explanation:
Each input is assigned to exactly one output.
Question 5.
_______________
Answer:
Not a function
Explanation:
The input value is 4 is paired with two outputs 25 and 35
Question 6.
The graph shows the relationship between the weights of 5 packages and the shipping charge for each package. Is the relationship represented by the graph a function? Explain.
_______________
Answer:
Function
Explanation:
Each input is assigned to exactly one output.
Essential Question Check-In
Independent Practice – Identifying and Representing Functions – Page No. 159
Determine whether each relationship represented by the ordered pairs is a function. Explain.
Question 8.
(2, 2), (3, 1), (5, 7), (8, 0), (9, 1)
_______________
Answer:
Function
Explanation:
Each input value is paired with exactly one output value.
Question 9.
(0, 4), (5, 1), (2, 8), (6, 3), (5, 9)
_______________
Answer:
Not a function
Explanation:
The input value is 5 is paired with two outputs 1 and 9
Question 10.
Draw Conclusions
Joaquin receives $0.40 per pound for 1 to 99 pounds of aluminum cans he recycles. He receives $0.50 per pound if he recycles more than 100 pounds. Is the amount of money Joaquin receives a function of the weight of the cans he recycles? Explain your reasoning.
_______________
Answer:
Yes
Explanation:
The amount of money increases with the weight of the cans. No weight will result in the same amount of money earned.
Question 11.
A biologist tracked the growth of a strain of bacteria, as shown in the graph.
a. Explain why the relationship represented by the graph is a function.
Type below:
_______________
Answer:
The relationship is a function as each input has been assigned exactly one output. There is only one number of bacteria for each number of hours.
Question 11.
b. What If?
Suppose there was the same number of bacteria for two consecutive hours. Would the graph still represent a function? Explain.
Type below:
_______________
Answer:
Yes. If the number of bacteria for two consecutive hours is the same, one input will still be paired with one output, hence the relationship is still a function.
Question 12.
Multiple Representations
Give an example of a function in everyday life, and represent it as a graph, a table, and a set of ordered pairs. Describe how you know it is a function.
Type below:
_______________
Answer:
The cost of a bouquet of flowers and the number of flowers in the bouquet is a functions. The unit cost of flowers = $0.85 and x the number of flowers. Hence, C= 0.85x
(2, 1.7), (4, 3.4), (6, 5.1), (8, 6.8), (10, 8.5)
Each value of the input is paired with exactly one output.
Identifying and Representing Functions – Page No. 160
The graph shows the relationship between the weights of six wedges of cheese and the price of each wedge.
H.O.T.
Focus on Higher Order Thinking
Question 15.
Justify Reasoning
A mapping diagram represents a relationship that contains three different input values and four different output values. Is the relationship a function? Explain your reasoning.
_______________
Answer:
No. Since there are three inputs and four outputs, one of the inputs will have more than one output, hence the relationship cannot be a function.
Question 16.
Communicate Mathematical Ideas
An onion farmer is hiring workers to help harvest the onions. He knows that the number of days it will take to harvest the onions is a function of the number of workers he hires. Explain the use of the word “function” in this context.
Type below:
_______________
Answer:
Number of days = f(number of workers)
Explanation:
We know that the more the number of workers will be involved in the harvesting of onion, the lesser days it will take to complete.
Thus the number of workers becomes the independent variable and the number of days becomes the dependent variable.
Here the word function is used to describe that the number of days is dependent on the number of workers.
Number of days = f(number of workers)
Guided Practice – Describing Functions – Page No. 164
Plot the ordered pairs from the table. Then graph the function represented by the ordered pairs and tell whether the function is linear or nonlinear.
Question 1.
y = 5 − 2x
_______________
Answer:
The graph of a linear function is a straight line
Linear relationship
Question 2.
y = 2 − x2
_______________
Answer:
y = 2 − x2
Graph the ordered pairs. Then draw a line through the points to represent the solution.
The graph of a linear function is not a straight line
Non-linear relationship
Explain whether each equation is a linear equation.
Question 3.
y = x2 – 1
_______________
Answer:
The equation is not in the form of a linear equation, hence is not linear equation.
Explanation:
Compare the equation with the general linear equation y = mx + b.
The equation is not in the form of a linear equation, hence is not linear equation.
Question 4.
y = 1 – x
_______________
Answer:
The equation is in the form of a linear equation, hence is a linear equation.
Explanation:
Compare the equation with the general linear equation y = mx + b.
The equation is in the form of a linear equation, hence is a linear equation.
Essential Question Check-In
Question 5.
Explain how you can use a table of values, an equation, and a graph to determine whether a function represents a proportional relationship.
Type below:
_______________
Answer:
From a table, determine the ratio y/x. If it is constant the relationship is proportional.
From a graph, note if the graph passes through the origin. The graph of a proportional relationship must pass through the origin (0, 0).
From an equation, compare with the general linear form of the equation, y = mx + b. If b = 0, the relationship is proportional.
Independent Practice – Describing Functions – Page No. 165
Question 6.
State whether the relationship between x and y in y = 4x – 5 is proportional or nonproportional. Then graph the function.
_______________
Answer:
Question 7.
The Fortaleza telescope in Brazil is a radio telescope. Its shape can be approximated with the equation y = 0.013x2. Is the relationship between x and y linear? Is it proportional? Explain.
____________
____________
Answer:
Compare the equation with the general linear equation y = mx + b.
The equation is not in the form of a linear equation, hence it is not a linear equation. Since x is squared, it is not proportional.
Question 8.
Kiley spent $20 on rides and snacks at the state fair. If x is the amount she spent on rides, and y is the amount she spent on snacks, the total amount she spent can be represented by the equation x + y = 20. Is the relationship between x and y linear? Is it proportional? Explain.
____________
____________
Answer:
x + y = 20
Rewriting the equation
y = 20 – x
Compare the equation with the general linear equation y = mx + b.
It is linear
Since b is not equal to 0, the relationship is not proportional.
Question 9.
Represent Real-World Problems
The drill team is buying new uniforms. The table shows y, the total cost in dollars, and x, the number of uniforms purchased.
a. Use the data to draw a graph. Is the relationship between x and y linear? Explain.
____________
Answer:
The graph of a linear relationship is a straight line.
x and y are linear.
Question 9.
b. Use your graph to predict the cost of purchasing 12 uniforms.
$ ________
Answer:
$720
Explanation:
The cost of 12 uniforms is $720
Question 10.
Marta, a whale calf in an aquarium, is fed a special milk formula. Her handler uses a graph to track the number of gallons of formula y the calf drinks in x hours. Is the relationship between x and y linear? Is it proportional? Explain.
____________
____________
Answer:
The relationship is linear
The relationship is proportional
Explanation:
As the data lies on a straight line, the relationship is linear
As the graph passes through the origin, the relationship is proportional
Describing Functions – Page No. 166
Question 11.
Critique Reasoning
A student claims that the equation y = 7 is not a linear equation because it does not have the form y=mx + b. Do you agree or disagree? Why?
____________
Answer:
Disagree; The equation can be written in the form y = mx + b Where m is 0. The graph of the solutions is a horizontal line.
Question 12.
Make a Prediction
Let x represent the number of hours you read a book and y represent the total number of pages you have read. You have already read 70 pages and can read 30 pages per hour. Write an equation relating x hours and y pages you read. Then predict the total number of pages you will have read after another 3 hours.
_______ pages
Answer:
160 pages
Explanation:
Let x represent the number of hours you read a book and y represent the total number of pages you have read. You have already read 70 pages and can read 30 pages per hour.
m = 30; b = 70 pages
y = 30x + 70
x = 3 hrs
y = 30(3) + 70 = 160
H.O.T.
Focus on Higher Order Thinking
Question 13.
Draw Conclusions
Rebecca draws a graph of a real-world relationship that turns out to be a set of unconnected points. Can the relationship be linear? Can it be proportional? Explain your reasoning.
Type below:
______________
Answer:
The relationship is linear if all the points lie on the same line. If the relationship is linear and passes through the origin, it is proportional.
Question 14.
Communicate Mathematical Ideas
Write a real-world problem involving a proportional relationship. Explain how you know the relationship is proportional.
Type below:
______________
Answer:
The amount of money earned at a car wash is a proportional relationship. When there are 0 cars washed, $0 are earned. The amount of money earned increases by the unit cost of a car wash.
Guided Practice – Comparing Functions – Page No. 170
Doctors have two methods of calculating maximum heart rate. With the first method, the maximum heart rate, y, in beats per minute is y = 220 − x, where x is the person’s age. The maximum heart rate with the second method is shown in the table.
Question 2.
Are heart rate and age proportional or nonproportional for each method?
____________
Answer:
For method 1, the relationship is non-proportional.
For method 2, the relationship is non-proportional.
Explanation:
Compare the equation with the general linear equation y = mx + b.
It is linear
Since b is not equal to 0, the relationship is not proportional.
Aisha runs a tutoring business. With Plan 1, students may choose to pay $15 per hour. With Plan 2, they may follow the plan shown on the graph.
Question 4.
Sketch a graph showing the $15 per hour option.
Type below:
______________
Answer:
Question 5.
What does the intersection of the two graphs mean?
Type below:
______________
Answer:
The intersection of the two graphs represents the number of hours for which both plans will cost the same,
Question 7.
Are cost and time proportional or nonproportional for each plan?
Type below:
______________
Answer:
Comparing with the general linear form of equation y = mx + b.
Since b = 0, the relationship is proportional
The cost and time are proportional for Plan 1
Comparing with the general linear form of equation y = mx + b.
Since b is not equal to 0, the relationship is proportional
The cost and time are not proportional for Plan 2
Essential Question Check-In
Question 8.
When using tables, graphs, and equations to compare functions, why do you find the equations for tables and graphs?
Type below:
______________
Answer:
The tables and graphs represent a part of the solution of the function. By writing the equation, any value can be a substitute to evaluate the function and compared it with the equations.
Independent Practice – Comparing Functions – Page No. 171
The table and graph show the miles driven and gas used for two scooters.
Question 10.
Are gas used and miles proportional or nonproportional for each scooter?
______________
Answer:
The gas used and miles are proportional to both scooters.
Explanation:
Compare with the general linear form of an equation, y = mx + b. If b = 0, the relationship is proportional.
The gas used and miles are proportional to both scooters.
A cell phone company offers two texting plans to its customers. The monthly cost, y dollars, of one plan is y = 0.10x + 5, where x is the number of texts. The cost of the other plan is shown in the table.
Question 12.
The graph of the first plan does not pass through the origin. What does this indicate?
Type below:
______________
Answer:
Plan 1
y = 0.10x + 5
The graph that does not pass through the origin indicates that there is a base price of $5 for the plan.
Question 13.
Brianna wants to buy a digital camera for a photography class. One store offers the camera for $50 down and a payment plan of $20 per month. The payment plan for a second store is described by y = 15x + 80, where y is the total cost in dollars and x is the number of months. Which camera is cheaper when the camera is paid off in 12 months? Explain.
______________
Answer:
For first store, the slope interecept form y = mx + b where m = 20 dollars per month and b = 50 dollar.
y = 20x + 50
x = 12 months
y = 20(12) + 50 = $290
Second store
y = 15x + 80
x = 12 months
y = 15(12) + 80 = $260
Compare the cost of camera if it paid off in 12 months $290 > $260
Camera is cheaper at second store
Comparing Functions – Page No. 172
Question 14.
The French club and soccer team are washing cars to earn money. The amount earned, y dollars, for washing x cars is a linear function. Which group makes the most money per car? Explain.
______________
Answer:
Soccer club makes the most money per car
H.O.T.
Focus on Higher Order Thinking
Question 15.
Draw Conclusions
Gym A charges $60 a month plus $5 per visit. The monthly cost at Gym B is represented by y = 5x + 40, where x is the number of visits per month. What conclusion can you draw about the monthly costs of gyms?
__________ is more expensive
Answer:
Gym A is more expensive
Explanation:
Since the rate per visit is the same, the monthly cost of Gyn A is always more than Gym B.
Question 17.
Analyze Relationships
The equations of two functions are y = −21x + 9 and y = −24x + 8. Which function is changing more quickly? Explain.
______________
Answer:
y = -21x + 9
y = -24x + 8
y = -24x + 8 is changing more quickly as the absolute value of -24 is greater than the absolute value of -21.
Guided Practice – Analyzing Graphs – Page No. 176
In a lab environment, colonies of bacteria follow a predictable pattern of growth. The graph shows this growth over time.
Question 1.
What is happening to the population during Phase 2?
______________
Answer:
For Phase 2, the graph is increasing quickly. This shows a period of rapid growth.
Question 2.
What is happening to the population during Phase 4?
______________
Answer:
In Phase 4, the graph is decreasing, hence the number of bacteria is decreasing.
The graphs give the speeds of three people who are riding snowmobiles. Tell which graph corresponds to each situation.
Question 3.
Chip begins his ride slowly but then stops to talk with some friends. After a few minutes, he continues his ride, gradually increasing his speed.
______________
Answer:
Graph 2
Explanation:
The slope of the graph is increasing, then it becomes constant and starts increasing again.
Graph 2
Question 4.
Linda steadily increases her speed through most of her ride. Then she slows down as she nears some trees.
______________
Answer:
Graph 3
Explanation:
The slope of the graph is increasing and then decreasing.
Graph 3
Question 5.
Paulo stood at the top of a diving board. He walked to the end of the board and then dove forward into the water. He plunged down below the surface, then swam straight forward while underwater. Finally, he swam forward and upward to the surface of the water. Draw a graph to represent Paulo’s elevation at different distances from the edge of the pool.
Type below:
______________
Answer:
Independent Practice – Analyzing Graphs – Page No. 177
Tell which graph corresponds to each situation below.
Question 6.
Arnold started from home and walked to a friend’s house. He stayed with his friend for a while and then walked to another friend’s house farther from home.
______________
Answer:
Graph 3
Explanation:
The graph increases (as Arnold walks from home to their friend’s house), then becomes constant (when he stayed with his friend), and then increases again (when he walks to another friend’s house farther away).
Graph 3
Question 7.
Francisco started from home and walked to the store. After shopping, he walked back home.
______________
Answer:
Graph 1
Explanation:
The graph increases (as Francisco walked from home to the store), becomes constant (when he shops), and then decreases (as he walked back home)
Graph 1
Question 8.
Celia walks to the library at a steady pace without stopping.
______________
Answer:
Graph 2
Explanation:
The graph increases at a constant rate (as Celia walks to the library without any stops)
Graph 2
Regina rented a motor scooter. The graph shows how far away she is from the rental site after each half-hour of riding.
Question 9.
Represent Real-World Problems
Use the graph to describe Regina’s trip. You can start the description like this: “Regina left the rental shop and rode for an hour…”
Type below:
______________
Answer:
Regina left the rental shop and rode for an hour. She rested for half an hour and then started back. After half an hour, she changed her mind and rode for another half an hour. She rests for half an hour. Then she started back and ranched the rental site in 2 hours.
Question 10.
Analyze Relationships
Determine during which half-hour Regina covered the greatest distance.
Type below:
______________
Answer:
Regina covered the greatest distance between 0.5 to 1hr of the journey. She covered 12 miles.
Analyzing Graphs – Page No. 178
The data in the table shows the speed of a ride at an amusement park at different times one afternoon.
Question 11.
Sketch a graph that shows the speed of the ride over time.
Type below:
______________
Answer:
Question 12.
Between which times is the ride’s speed increasing the fastest?
Type below:
______________
Answer:
The speed is increasing the fastest during the 3: 21 and 3: 22
Question 13.
Between which times is the ride’s speed decreasing the fastest?
Type below:
______________
Answer:
The speed is decreasing the fastest during the 3: 23 and 3: 24
H.O.T.
Focus on Higher Order Thinking
Question 14.
Justify Reasoning
What is happening to the fox population before time t? Explain your reasoning.
Type below:
______________
Answer:
The population decreases and then increases before time t
Question 15.
What If?
Suppose at time t, a conservation organization moves a large group of foxes to the island. Sketch a graph to show how this action might affect the population on the island over time t.
Type below:
______________
Answer:
Explanation:
The population is decreasing at first, then it is increasing rapidly.
Question 16.
Make a Prediction
At some point in time t, a forest fire destroys part of the woodland area on the island. Describe how your graph from problem 15 might change.
Type below:
______________
Answer:
The population would dramatically decrease if there was a fire due to a lack of food supply and good land.
6.1 Identifying and Representing Functions – Model Quiz – Page No. 179
Determine whether each relationship is a function.
Question 1.
__________
Answer:
Not a function
Explanation:
A relationship is a function when each input is paired with exactly one output. The input 5 has more than one output.
Not a function
Question 2.
__________
Answer:
Function
Explanation:
A relationship is a function when each input is paired with exactly one output.
Each input is paired with only one output.
Function
Question 3.
(2, 5), (7, 2), (−3, 4), (2, 9), (1, 1)
__________
Answer:
Not a function
Explanation:
A relationship is a function when each input is paired with exactly one output. Input 2 has more than one output.
Not a function
6.2 Describing Functions
Determine whether each situation is linear or nonlinear, and proportional or nonproportional.
Question 4.
Joanna is paid $14 per hour.
__________
__________
Answer:
Linear
Proportional
Explanation:
Writing the situation as an equation, where x is the number of hours.
y = 14x
Compare with the general linear equation y = mx + b
Linear
Since b = 0, the relationship is proportional.
Proportional
Question 5.
Alberto started out bench pressing 50 pounds. He then added 5 pounds every week.
__________
__________
Answer:
Linear
Non-proportional
Explanation:
Writing the situation as an equation, where x is the number of hours.
y = 5x + 50
Compare with the general linear equation y = mx + b
Linear
Since b is not equal to 0, the relationship is non-proportional.
Non-proportional
6.3 Comparing Functions
Question 6.
Which function is changing more quickly? Explain.
__________
Answer:
Function 2 is changing more quickly.
6.4 Analyzing Graphs
Question 7.
Describe a graph that shows Sam running at a constant rate.
Type below:
______________
Answer:
The graph would be a straight line
Explanation:
Since Sam is running at a constant rate, the distance covered per unit of time remains the same and the relationship is linear and proportional.
The graph would be a straight line
Essential Question
Question 8.
How can you use functions to solve real-world problems?
Type below:
______________
Answer:
If in the equation the power of x is 1 then it is linear otherwise nonlinear.
In a graph, if the points form a line it is linear if they form a curve it is a nonlinear function.
Selected Response – Mixed Review – Page No. 180
Question 1.
Which table shows a proportional function?
Options:
a. A
b. B
c. C
d. D
Answer:
c. C
Explanation:
It contains the ordered pair of the origin (0, 0)
Option C represents a proportional relationship.
Question 2.
What is the slope and y-intercept of the function shown in the table?
Options:
a. m = -2; b = -4
b. m = -2; b = 4
c. m = 2; b = 4
d. m = 4; b = 2
Answer:
c. m = 2; b = 4
Explanation:
Find the slope using two points from the grapgh by
Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (1, 6) and (x2, y2) = (4, 12)
Slope m = (y2 -y1)/(x2 – x1) = (12 – 6)/(4 – 1) = 6/3 = 2
Substituting the value of the slope m and (x, y) to find the slope-intercept form.
12 = 4(2) + b
y-intercept b = 4
Question 3.
The table below shows some input and output values of a function.
What is the missing output value?
Options:
a. 20
b. 21
c. 22
d. 23
Answer:
b. 21
Explanation:
Find the rate of change = (17.5 – 14)/(5 – 4) = 3.5
Since the missing output is corresponding to x = 6 and 3.5 to 17.5 (for x = 5)
Output = 17.5 + 3.5 = 21
Explanation:
The graph would increase at a constant rate and would decrease at a constant rate.
The graph would be the upside-down V-shaped
Mini-Task
Question 5.
Linear functions can be used to find the price of a building based on its floor area. Below are two of these functions.
y = 40x + 15,000
a. Find and compare the slopes.
Type below:
____________
Answer:
Compare the slopes
The slope for the first function is less than the slope of the second function.
y = 40x + 15000
Compare with slope intercept form y = mx + b where m is the slope m = 40
Second function find the slope using given points by Slope m = (y2 -y1)/(x2 – x1) where (x1, y1) = (7, 3) and (x2, y2) = (6, 4)
Slope m = (y2 -y1)/(x2 – x1) = (56000 – 32000)/(700 – 400) = 24000/300 = 80
m = 80
Question 5.
b. Find and compare the y-intercepts.
Type below:
____________
Answer:
y = 40x + 15,000
Compare with slope-intercept form y = mx + b where m is the slope b = 15000
The second function finds the slope using given points by Slope m and (x, y) in the slope-intercept form to find y-intercept b
y = mx + b where (x, y) = (700, 56000) and m = 80
56000 = 80(700) + b
b = 0
Compare y-intercepts
The y-intercept of the first function is greater than the y-intercept of the second function
Question 5.
c. Describe each function as proportional or nonproportional.
Type below:
____________
Answer:
Comparable to slope intercept form y = mx + b
First function: y = 40x + 15000
The second function: y = 80x
Since b is not equal to 0
The first function is non-proportional
Since b = 0
The second function is proportional.
Conclusion:
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