Students can use the Spectrum Math Grade 8 Answer Key Chapter 4 Pretest as a quick guide to resolve any of their doubts.

Check What You Know

Functions

Decide if each table represents a function by stating yes or no.

Question 1.
a.

______
No,

Explanation:
This table does not represent a function because one of the input variables has more than one output variable.
In out put two variables are there for single input.

b.

______
Yes,

Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
output = input + b
-4 = 6 + b
-3 = 2 + b
1 = 0 + b
7 = 2 + b

c.

______
Yes,

Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
-3 = 4 + b
-2 = 5 + b
0 = 0 + b
4 = 8 + b

Complete each function table for the given function.

Question 2.
a. y = 18x – 4

Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
y = 18x – 4
out put = input – 4
y = 18(-15) – 4 = -270 – 4 = -274
y = 18(-11) – 4 = -198 – 4 = -202
y = 18(5) – 4 = 90 – 4 = 86
y = 18(9) – 4 = 162 – 4 = 158
y = 18(12) – 4 = 216 – 4 = 212

b. y = x – 19

Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
y = x – 19
out put = input – 19
y = -63 – 19 = -82
y = -42 – 19 = – 61
y = -28 – 19 = -47
y = 37 – 19 = 18
y = 55 – 19 = 36

c. y = 12x + 3

Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
y = 12x + 3
out put = input + 3
y = 12(-13) + 3 = -156 + 3 = -153
y = 12(-6) + 3 = -72 + 3 = -69
y = 12(-4) + 3 = -48 + 3 = -45
y = 12(10) + 3 = 120 + 3 = 123
y = 12(12) + 3 = 144 + 3 = 147

Find the relationship for each function table and then complete the table.

Question 3.
a.

Function: _____
31,  85, y = 9x + 4

Explanation:
The table of values represents a function.
A linear relationship is in the form of y = mx + b exists.
y = mx + b
Step 1: Find the rate of change by calculating the slope, or rate of change, between the two variables.
$$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
m = $$\frac{94-40}{10-4}$$ = $$\frac{54}{6}$$ = 9
Step 2: Substitute known values of x and y with the slope into the formula y = mx + b.
y = mx + b
94 = 9 x 10 + b
94 – 90 = b
b = 4
Step 3: Use the found values in the linear function to complete the table.
y = mx + b.
y = 9 (3) + 4
for x = 3 :: y = 31
y = mx + b.
y = 9 (9) + 4
for x = 9 :: y = 85
So, the relationship of the function table is y = 9x + 4
The value of y = 31, 85

b.

Function: _____
4, 5, y = $$\frac{1}{7}$$x – 5

Explanation:
The table of values represents a function.
A linear relationship is in the form of y = mx + b exists.
y = mx + b
Step 1: Find the rate of change by calculating the slope, or rate of change, between the two variables.
$$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
m = $$\frac{6-(-3)}{77-14}$$ = $$\frac{9}{63}$$ = $$\frac{1}{7}$$
Step 2: Substitute known values of x and y with the slope into the formula y = mx + b.
y = mx + b
-3 = $$\frac{1}{7}$$ x 14 + b
-3 -2 = b
b = -5
Step 3: Use the found values in the linear function to complete the table.
y = mx + b.
y = $$\frac{1}{7}$$ (63) – 5
y = 9 – 5 = 4

for x = 63 :: y = 4
y = mx + b.
y = $$\frac{1}{7}$$ (70) – 5
y = 10 – 5 = 5
for x = 70 :: y = 5
So, the relationship of the function table is y = 9x + 4
The value of y = 4, 5

Find the rate of change, or slope, for points on the function table and decide if it represents a linear or nonlinear relationship.

Question 4.
a.

Relationship:
___________
Nonlinear.

Explanation:
The rate of change in a function table by using the slope formula,
$$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$, across multiple points on the table.
Slope (m) = $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
Rate = $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
Rate =  $$\frac{10-14}{2 – 0}$$
Rate =  $$\frac{-4}{2}$$
Rate = -2

Rate = $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
Rate =  $$\frac{12-8}{5 – 3}$$
Rate =  $$\frac{4}{2}$$
Rate = 2

b.

Relationship:
___________
Linear.

Explanation:
The rate of change in a function table by using the slope formula,
$$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$, across multiple points on the table.
Slope (m) = $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
Rate = $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
Rate =  $$\frac{-11-(-17)}{-1 – (-2)}$$
Rate =  $$\frac{6}{1}$$

Rate = $$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
Rate =  $$\frac{1-(-5)}{1 – 0}$$
Rate =  $$\frac{6}{1}$$
Rate = 6

Find the rate of change for each function table. Write fractions in simplest form.

Question 5.
a.

rate of change:
___________
1

Explanation:
The rate of change that exists in a function can be calculated by finding the ratio of the amount of change in the output variable to the amount of change in the input variable.

$$\frac{17 – 8}{12 – 3}$$
= $$\frac{9}{9}$$ = 1

b.

rate of change:
___________
$$\frac{1}{9}$$

Explanation:
The rate of change that exists in a function can be calculated by finding the ratio of the amount of change in the output variable to the amount of change in the input variable.

$$\frac{10 – 3}{90 – 27}$$
= $$\frac{7}{63}$$
= $$\frac{1}{9}$$

c.

rate of change:
___________
$$\frac{5}{4}$$

Explanation:
The rate of change that exists in a function can be calculated by finding the ratio of the amount of change in the output variable to the amount of change in the input variable.

$$\frac{12 – 2}{14 – 6}$$
= $$\frac{10}{8}$$
= $$\frac{5}{4}$$

Find the initial value of each function represented below.

Question 6.
a.

b = ____
-1

Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = $$\frac{19 – 3}{10 – 2}$$
= $$\frac{16}{8}$$
= 2
y = mx + b,
for x = 2, y = 3 the initial value b  is,
3 = 2(2) + b
3 – 4 = b
b = -1

b.

b = ____
2

Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = $$\frac{18 – 2}{4 – 0}$$
= $$\frac{16}{4}$$
= 4
y = mx + b,
for x = 0, y = 2 the initial value b  is,
2 = 4(0) + b
b = 2

c.

b = ____
3

Explanation:
The initial value of a linear function is to solve the equation for when the input, or x, equals 0. Use the formula y = mx + b,
where m represents the rate of change and b represents the initial value of the linear.
m = $$\frac{11 – 3}{4 – 0}$$
= $$\frac{8}{4}$$
= 2
y = mx + b,
for x = 0, y = 3 the initial value b  is,
3 = 2(0) + b
b = 3

Use the information given to find the function models for the linear functions shown.

Question 7.
a. (4, 2) and (8, 5)
y = _____
y = $$\frac{3}{4}$$x – 1

Explanation:
m = $$\frac{5 – 2}{8 – 4}$$
m = $$\frac{3}{4}$$
for a point (4, 2) initial value
y = mx + b
2 = $$\frac{3}{4}$$4 + b
b = 2 – 3
b = -1
y = $$\frac{3}{4}$$x – 1

b.

y = _____
y = $$\frac{1}{6}$$x

Explanation:
To find the rate of change by calculating the slope, or rate of change, between the two variables.
$$\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$$
$$\frac{5-1}{30-6}$$ = $$\frac{4}{24}$$
y = = $$\frac{1}{6}$$

c.

y = _____
y = $$\frac{1}{3}$$x + 4

Explanation:
for x= 0 initial value b
y = 4 as per the graph.
b = 4
Mark the point where the line will cross the y-axis (b = 4).
Draw a line that goes directly through the points found.
(-3, 3) and (3, 5)
m = $$\frac{5-1}{3-(3)}$$
m = $$\frac{4}{6}$$
m = $$\frac{2}{3}$$
y = $$\frac{1}{3}$$x + 4

Compare the rate of change for the equation and table and decide which has a greater rate of change by writing equation or table.

Question 8.
y = 6x – 2 or
____________
Rate of change for table = $$\frac{-13-5}{6-(-3)}$$ = $$\frac{18}{9}$$ = 2