Average Rate of Change
Name_______ANSWERS_______________
Directions: Find the average rate of change for the following problems. Show your work.
The average rate of change of a function y = f (x) over the interval [a,b] is
f (b )  f ( a )
. ba
The average rate of change of a function corresponds to the slope of the line segment
(called the secant line) connecting the two endpoints of the given interval.
1. a. Graph: f (x) = x² + x – 2.
b. Plot the points (2,4) and (0,-2) on the graph.
c. Draw a line segment connecting these two points.
d. What is the slope of this line segment? _6/2 = 3_____
e. Using the formula for “average rate of change”, find the
average rate of change on the interval from x = 0 to x = 2.
f (0)  f (2) 2  4 6


 3 02
2
2
2. Given:
x
y
-2
-8
-1
-1
0
0
1
1
2
8
a. Determine the average rate of change of y over the interval from x = -1 to x = 1. f ( 1)  f (1) 1  1 2


1
1  1
2
2
b. Determine the average rate of change of y over the interval -2 ≤ x ≤ 1. f ( 2)  f (1) 8  1 9


3
2  1
3
3
c. Determine the average rate of change of y over the interval [-2, 2]. f ( 2)  f (2) 8  8 16


4
2  2
4
4
3. Calculate the average rate of change of the function over the given interval.
f (2)  f (5) 11  23 12
f (x) = 4x + 3 over the interval [2, 5] 

 4 3
3
25
Since this function is linear, the average rate of change is the slope of the line.
All Rights Reserved © MathBits.com 4. Calculate the average rate of change in library income between 2010 and 2013.
Library Income
in Dollars
14,587
15,678
16,988
18,389
19,089
20,870
Year
2008
2009
2010
2011
2012
2013
f (2010)  f (2013) 16988  20870 3882


 1294
2010  2013
3
3
$1294.00
5. When the average rate of change of a function is constant, the function is linear. Determine if the
following function is linear by examining the average rates of change. Explain your decision.
f (3)  f (4) 6  8 14


 2 This function is linear: f (x) = ‐2x 3  4
7
7
f (1)  f (4) 2  8 10
f (0)  f (4) 0  8 8
f (3)  f (0) 6  0 6


 2 

 2


 2 4
4
3  0
3
3
04
1  4
5
5
x
f (x)
-3
6
-1
2
0
0
4
-8
6. Given f (x) = 2x². Find a value b such that the average rate of change of f (x) from x = 1 to x = b
equals 12. f (1)  f (b) 2  2b 2

 12
1 b
1 b
2
2  2b  12  12b
b = 5
0  2b  12b  10
2
0  b 2  6b  5; 0  (b  5)(b  1); b  5, b  1
7. In 1997, a town of 50,000 people started growing by 1,000 people each year. Complete the table
showing the town’s population every 5 years over a 15-year period.
Year
1997
2002
2007
2012
Population
50,000
55,000
60,000
65,000
a. What is the average rate of change over the 15-year period? f (1997)  f (2012) 50000  65000 15000


 1000 15
15
1997  2012
b. Is the average rate of change constant? _yes__ What does this
tell you about the type of function? ___linear______________
c. Write a formula for the population, P, as a function of x, in years. P = 1000x + 50000
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