Students can use the Spectrum Math Grade 8 Answer Key Chapter 4 Posttest as a quick guide to resolve any of their doubts.
Spectrum Math Grade 8 Chapter 4 Posttest Answers Key
Check What You Learned
Functions
Decide if each table represents a function by stating yes or no.
Question 1.
a.
_________
Answer:
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.
_________
Answer:
Yes,
Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
output = input + b
-1 = 2 + b
0 = 2 + b
1 = -3 + b
2 = -2 + b
c.
_________
Answer:
Yes,
Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
output = input + b
-3 = 4 + b
-2 = 4 + b
-1 = -1 + b
3 = -1 + b
Complete each function table for the given function.
Question 2.
a. 3x + 24
Answer:
Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
out put = input + 24
y = 3x + 24
y = 3(10) + 24 = 54
y = 3(14) + 24 = 66
y = 3(19) + 24 = 81
y = 3(25) + 24 = 99
y = 3(29) + 24 = 111
b. y = 2x – 13
Answer:
Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
out put = input – 13
y = 2x – 13
y = 2(15) – 13 = 17
y = 2(29) – 13 = 45
y = 2(37) – 13 = 61
y = 2(59) – 13 = 105
y = 2(73) – 13 = 133
c. y = \(\frac{1}{19}\)x + 3
Answer:
Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
out put = input + 3
y = \(\frac{1}{19}\)x + 3
y = \(\frac{1}{19}\) 38 + 3 = 2 + 3 = 5
y = \(\frac{1}{19}\) 76 + 3 = 4 + 3 = 7
y = \(\frac{1}{19}\) 171 + 3 = 9 + 3 = 12
y = \(\frac{1}{19}\) 228 + 3 = 12 + 3 = 15
y = \(\frac{1}{19}\) 285 + 3 = 15 + 3 = 18
Find the relationship for each function table and then complete the table.
Question 3.
a.
Function: ______
Answer:
0, 6; y = x – 6
Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
out put = input – 6
y = x – 6
y = 6 – 6 = 0
for x = 7
y = 7 – 6 = 1
for x = 8
y = 8 – 6 = 2
for x = 12
y = 12 – 6 = 6
b.
Function: ______
Answer:
0, -2; y = -x + 7
Explanation:
A function is a relationship between two variables which results in only one output value for each input value.
out put = input + 7
y = -x + 7
y = -1 + 7 = 6
y = -7 + 7 = 0
y = -9 + 7 = -2
y = -11 + 7 = -4
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:
_________
Answer:
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 = \(\frac{-7-(-3)}{-2 – (-1)}\)
Rate = \(\frac{-4}{-1}\)
Rate = 4
Rate = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Rate = \(\frac{5-1}{1 – 0}\)
Rate = \(\frac{4}{1}\)
Rate = 4
b.
Relationship:
_________
Answer:
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 = \(\frac{2-0}{1 – 0}\)
Rate = \(\frac{2}{1}\)
Rate = 2
Rate = \(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
Rate = \(\frac{26-8}{3 – 2}\)
Rate = \(\frac{18}{1}\)
Rate = 8
Find the rate of change for each function table Write fractions in simplest form.
Question 5.
a.
rate of change:
____________
Answer:
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.
\(\frac{19 – 3}{6 – 2}\)
= \(\frac{16}{4}\) = 4
b.
rate of change:
____________
Answer:
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.
\(\frac{26 – 10}{6 – 2}\)
= \(\frac{16}{4}\) = 4
c.
rate of change:
____________
Answer:
\(\frac{1}{2}\)
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.
\(\frac{4 – 0}{11 – 3}\)
= \(\frac{4}{8}\)
= \(\frac{1}{2}\)
Find the initial value of the function represented below.
Question 6.
Jacob is filling a flower bed with dirt. It already has 5 cubic ft. of dirt in it. After 30 minutes of shoveling, the flower bed has 20 cubic ft. in it. How much dirt will be in the flower bed after 1 hour?
Initial Value: _______
Answer:
Initial value is 5.
Explanation:
Jacob is filling a flower bed with dirt 5 cubic ft. of dirt in it.
After 30 minutes of shoveling, the flower bed has 20 cubic ft. in it.
Total dirt will be in the flower bed after 1 hour.
y = mx + b
It already has 5 cubic ft
it means, b = 5 initial value.
y = mx + 5
Use the given information to find the function models for the linear functions shown.
Question 7.
a.
(4, 5) and (5, 8)
y = _____
Answer:
y = 3x – 7
Explanation:
m = \(\frac{8 – 5}{5 – 4}\)
m = \(\frac{3}{1}\)
m = 3
for a point (4, 5) initial value
y = mx + b
5 = 3 (4) + b
b = 5 – 12
b = -7
y = 3x – 7
b.
y = _____
Answer:
y = 10x + 6
Explanation:
To find the rate of change by calculating the slope, or rate of change, between the two variables.
\(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\)
\(\frac{26-6}{2-0}\) = \(\frac{20}{2}\)
y = 10
A function is a relationship between two variables which results in only one output value for each input value.
y = 10x + 6
out put = input + 6
6 = 10(0) + 6
16 = 10(1) + 6
26 = 10(2) + 6
c.
y = ____
Answer:
y = –\(\frac{2}{3}\)x + 3
Explanation:
for x= 0
y = –\(\frac{2}{3}\)x + 3
y = –\(\frac{2}{3}\)(0) + 3
y = 3
Mark the point where the line will cross the y-axis (b = 3).
Draw a line that goes directly through the points found.
Sketch the linear function shown below.
Question 8.
y = \(\frac{1}{2}\)x – 2
Answer:
Explanation:
for x= 0
y = \(\frac{1}{2}\)x – 2
y = -2
Mark the point where the line will cross the y-axis (b = -2).
Draw a line that goes directly through the points found.
y = \(\frac{1}{2}\)x – 2
y = -2
for x= 2
y = \(\frac{2}{2}\) – 2
y = -1
y = \(\frac{1}{2}\)4 – 2
for x= 4
y = 0