8th Math functions
Linear vs. Nonlinear
1. Which of the following best describes the equation below?
y = 3x + 10
A. both linear and nonlinear
B. nonlinear
C. neither linear nor nonlinear
D. linear
Functions
2. Which of the following is the graph of the function shown below?
y = x2 + 4x + 3
W.
X.
Y.
Z.
A. Z
B. X
C. W
D. Y
Functions
3. Which of the following relations is not a function?
A.
B.
C.
D.
(-6, 4), (2, 3), (-5, 1), (8, 2)
(-5, 4), (2, 2), (8, 1), (-6, 5)
(8, 4), (-5, 2), (2, 1), (-6, 2)
(2, 4), (-5, 2), (2, 1), (-6, 2)
Functions
4. Determine if the following functions are increasing or decreasing, and compare their rates of
change.
A.
B.
C.
D.
Both functions are decreasing and have different rates of change.
Both functions are decreasing and have the same rate of change.
Both functions are increasing and have the same rate of change.
Both functions are increasing and have different rates of change.
Linear vs. Nonlinear
5. Holly was conducting a science experiment to determine the rate of cell division. The cells
that she was studying divided at a nonlinear rate. Which of the following graphs could represent
her results?
Y.
A. Graph Y and Graph Z
B. Graph Y
C. Graph Z
D. neither Graph Y nor Graph Z
Z.
Functions
6. Which of the following best describes the equation below?
y = |x| + 5
A. both a relation and a function
B. relation only
C. neither a relation nor a function
D. function only
Functions
7. Compare the y-intercepts and the rates of change of the following items.
A.
B.
C.
D.
The rates of change are the same, but the y-intercepts are different.
The items have the same y-intercept and the same rate of change.
The y-intercepts are the same, but the rates of change are different.
The items have different y-intercepts and different rates of change.
Linear vs. Nonlinear
8. Which of the following best describes the equation below?
y = 3x2 + 6
A. both linear and nonlinear
B. nonlinear
C. linear
D. neither linear nor nonlinear
Linear Relationships
9. Georgia filled her kitchen sink up with water so that she could do the dishes. When she was
done with the dishes, she pulled out the drain stopper so the water could begin to drain out of the
sink.
A linear model of this situation contains the values (1, 6.8) and (3, 6.4), where x represents the
number of seconds, and y represents the water level in the sink, in inches.
What is the rate of change in this linear model?
A. 0.2 of an inch per second
B. -0.2 of an inch per second
C. -0.4 of an inch per second
D. -7 inches per second
Linear Relationships
10. Carrie's Catering is having a special on their box lunches. The minimum purchase is 11
boxes for $88. Each additional box is $6. If x represents the number of boxes over the minimum,
which of the following equations represents the cost of box lunches during this special?
A. y = 6x + 88
B. y = 88x + 6
C. y = 88 - 6x
D. y = 88x - 6
Linear vs. Nonlinear
11.
The graph above represents what type of function?
A. nonlinear
B. both linear and nonlinear
C. linear
D. neither linear nor nonlinear
Linear vs. Nonlinear
12.
x
y
1
9
2
11
3
13
4
15
5
17
The table above represents what type of function?
A. linear and nonlinear
B. linear
C. neither linear nor nonlinear
D. nonlinear
Linear Relationships
13. A local animal shelter started the week with 45 kittens available for adoption. The rate of
adoptions of kittens was consistent for the first three days of the week. After those three days, 36
kittens were still available for adoption.
In a linear model of this situation, which of the following statements applies?
A.
B.
C.
D.
An additional day of adoptions is associated with an additional 1 kitten adopted.
An additional day of adoptions is associated with an additional 3 kittens adopted.
An additional day of adoptions is associated with an additional 6 kittens adopted.
An additional day of adoptions is associated with an additional 9 kittens adopted.
Linear Relationships
14. Ellen made pudding for dessert. She collected data on the temperature of the pudding as she
was cooking, as graphed below.
What was the original temperature of the pudding?
A. 70°F
B. 30°F
C. 3°F
D. 60°F
Linear Relationships
15. Julie bought a drink for $2.99, two bags of chips for $1.29 each, and some gasoline at a
convenience store. The price of gasoline was $2.59 per gallon.
If x represents the number of gallons of gasoline she purchased, which of the following equations
can be used to find how much Julie spent at the convenience store?
A. y = 5.57x + 2.59
B. y = 2.59x + 4.28
C. y = 4.28x + 2.59
D. y = 2.59x + 5.57
Answers
1. D
2. B
3. D
4. D
5. C
6. A
7. B
8. B
9. B
10. A
11. A
12. B
13. B
14. D
15. D
Explanations
1. A linear equation has a constant slope. A nonlinear equation does not have a constant slope.
Because the given equation has a constant slope, it is a linear equation.
2. Find the coordinates of points on the graph of the function by substituting x-values into the
equation and solving for y.
x = -2
x = -1
x=0
y = (-2)2 + 4(-2) + 3
= 4 + (-8) + 3
= -1
y = (-1)2 + 4(-1) + 3
= 1 + (-4) + 3
=0
y = (0)2 + 4(0) + 3
=0+0+3
=3
So, the graph of the function contains the points (-2, -1), (-1, 0), and (0, 3).
Therefore, graph X is the graph of the function y = x2 + 4x + 3.
3. A relation is a set of one or more ordered pairs.
A function is a relation in which each element of the domain is paired with exactly one element
of the range.
In other words, a function does not have two or more y-coordinates for any x-coordinate.
The only answer choice not meeting this requirement is (2, 4), (-5, 2), (2, 1), (-6, 2).
4. To determine the rate of change of the line described in item I, use the two given points and
the formula for slope.
Since the rate of change is positive, the function is increasing.
To determine the rate of change of the function represented by the table, choose two points and
use the formula for slope. This calculation uses (0, -4) and (4, -2).
Since the rate of change is positive, the function is increasing.
Therefore, both functions are increasing and have different rates of change.
5. The slope of the graphs represent the rate at which the cells divide.
Graph Z does not have a constant slope, so it represents a nonlinear function.
The other graph has a constant slope, so it represents a linear function.
6. A relation is a set of one or more ordered pairs.
A function is a relation in which each element of the domain is paired with exactly one element
of the range.
The graph of this equation has one y-coordinate for every x-coordinate.
This can be determined by using the vertical line test. A relation is a function if no vertical lines
can be drawn that cross the graph in more than one place. Since a vertical line will not cross the
graph twice anywhere, the relation is a function.
Therefore, the equation is best described as both a relation and a function.
7. Item I is an equation in the form y = mx + b, where m is the slope, or rate of change, and b
represents the y-intercept.
The rate of change of item I is
, and the y-intercept is (0, -1).
To determine the rate of change of the graph, choose two points and use the formula for slope.
This calculation uses (0, -1) and (2, 0).
The y-intercept of the graph is the point at which the line crosses the y-axis, (0, -1).
Therefore, the items have the same y-intercept and the same rate of change.
8. A linear equation has a constant slope. A nonlinear equation does not have a constant slope.
Because the given equation does not have a constant slope, it is a nonlinear equation.
9. The rate of change, or slope, can be determined from the two given points.
Therefore, the rate of change is -0.2 inches per second.
10. The cost of each additional box lunch, $6, indicates the rate of change.
Since the order increases by that amount for every box lunch over the minimum, and x represents
the number of boxes over the minimum, 6 is the value to be attached to the variable x.
Further, the cost of the additional boxes has to be added to the minimum purchase in order to
determine the total cost of the order.
Therefore, the equation which represents the total cost of the order is y = 6x + 88.
11. The graph of a linear function has a slope that remains constant. The graph of a nonlinear
function has a slope that does not remain constant.
The line on this graph is not straight. Therefore, its slope is not constant.
So, it represents a nonlinear function.
12. In a linear function, numbers increase at a constant rate.
Each number in the y column is two more than the previous number. The numbers are 2(1) + 7,
2(2) + 7, 2(3) + 7, 2(4) + 7, and 2(5) + 7.
So, the numbers are given by the linear equation y = 2x + 7.
13. In a linear model of this situation, the y-intercept represents the number of kittens available
for adoption at the beginning of the week, 45.
The rate of change, or slope, can be determined using two values: kittens available at start of
week, (0, 45), and kittens available after 3 days, (3, 36).
The linear model of this situation is y = 45 - 3x, where x represents the number of days of
adoptions.
The rate of change, or slope, indicates that an additional day of adoptions is associated with
an additional 3 kittens adopted.
14. To determine the original temperature of the pudding, identify the y-intercept of the graph.
Therefore, the original temperature was 60°F.
15. Julie bought a drink for $2.99 and two bags of chips for $1.29 each. So, she spent a total of
$5.57 on a drink and chips.
Each gallon of gasoline, represented by x, costs $2.59.
Therefore, the equation that can be used to find how much Julie spent at the convenience store is
shown below.
y = 2.59x + 5.57