Average Rate of Change WKST
1. Find the average rate of change over the indicated interval.
!1 $
b) f ( x ) = x 3 − x + 2 , # ,1&
"2 %
2. Sketch each function. Then determine if the average rate of change over the indicated interval is
positive or negative. Explain.
a) f ( x ) = −5x + 7 , [−2, 3]
b) f ( x ) = 3x−4 −1 , [-3, -1]
a) f ( x ) = − x + 2 , [1, 9]
3. The graph of f ( x ) is shown below. Over which interval does f ( x ) have an average rate of -4?
a) −1 < x < 1
b) −5 < x < 1
c) −2 < x < 4
d) −8 < x < 2
4. Over which interval does g ( x ) have a positive average rate of change?
x
g ( x)
-5
0
-4
2
-3
6
-2
7
-1
5
0
3
1
0
2
-3
3
-5
4
-1
5
2
5. Below is the graph and table for 2 runners running the 400-meter hurdles race.
A a)
b)
c)
d)
B Which runner has a faster average speed from 9 to 20 seconds?
Which runner has a faster average speed from 20 to 31 seconds?
Which runner has a faster average speed from 9 to 31 seconds?
Which runner wins the race? How do you know?
a) [-4,3]
b) [-3, 1]
c) [-2, 3]
d) [4, 5]
6. Suppose 25 flour beetles are left undisturbed in a warehouse bin. The beetle population doubles in
size every week. The equation P( x) = 25 • 2 x can be used to determine the number of beetles after
x weeks. Complete the table.
a) Calculate the average growth rate between weeks 1 and 3.
b) Calculate the average growth rate for the first five weeks
[0, 5].
c) Which average growth rate is higher? Why do you think it is
higher?
7. Given the graph m(x) and table n(x) below, compare the quantities in the table and decide if the
quantities are less than, greater than, or equal.
Graph m ( x )
x
6
3
0
-3
-6
First Quantity
𝑚 −3
Comparison < 𝑜𝑟 >
𝑜𝑟 =
n(x)
0
-2
-4
-6
-8
Second Quantity
𝑛 −3
1
2
𝑛 1
𝑚 8
𝑛 8
𝑚 0 − 𝑚 −3
0 − −3
𝑛 0 − 𝑛 −3
0 − −3
𝑚 0 − 𝑚 −1
0 − −1
𝑚 −1 − 𝑚 −2
−1 − −2
𝑛 3 −𝑛 0
3−0
The rate of change for the
function m(x) on the
interval −20 ≤ 𝑥 ≤ −19
𝑛 6 −𝑛 3
6−3
The rate of change for the
function n(x) on the
interval −20 ≤ 𝑥 ≤ −19
𝑚
8. Let f ( x ) = mx + b . Find the average rate of change of f ( x ) between the points x = a and x = a + h .
ANSWERS
1.
a) -5
b) ¾
2.
a) The average rate of change is negative because
the graph of f ( x ) is decreasing on the indicated
interval.
b) The average rate of change is
positive because the graph of f ( x ) is
increasing on the indicated interval.
3. A
4. D
5.
a) Runner B
b) Runner A
6.
a) 75 beetles/week
c) Neither
b) 31 beetles/week
d) Runner A. Runner A runs 400 meters
in less time than RunnerB.
c) The average growth is larger on the
interval [1,3] because P ( x ) increases
faster on the interval [1,3] than it does
on [0,5].
7.
Answer to
First
Quantity
-6
≈ 𝟐. 𝟕𝟓
negative
6
1
𝟐
𝟑
large &
positive
8. m
First
Quantity
𝑚 −3
1
𝑚
2
𝑚 8
𝑚 0 − 𝑚 −3
0 − −3
𝑚 0 − 𝑚 −1
0 − −1
𝑛 3 −𝑛 0
3−0
The rate of
change for the
function m(x)
on the interval
−20 ≤ 𝑥 ≤ −19
Comparison
< 𝑜𝑟 >
𝑜𝑟 =
=
Second
Quantity
𝑛 −3
>
𝑛 1
<
𝑛 8
𝑛 0 − 𝑛 −3
0 − −3
𝑚 −1 − 𝑚 −2
−1 − −2
𝑛 6 −𝑛 3
6−3
The rate of
change for the
function n(x)
on the interval
−20 ≤ 𝑥 ≤ −19
>
<
=
>
Answer to
Second
Quantity
-6
−𝟗
≈ −𝟑. 𝟑
𝟑
positive
𝟐
𝟑
3
𝟐
𝟑
𝟐
𝟑