1. Section 1.5
Functions and Change

2. Change
The change from A to B is the difference between A and B and is found by subtracting A from B.
Change from A to B is given by
B - A

3. Example 1 Finding Change
Find the change in temperature if the temperature went from F to F.

4. Relative Change
Relative change is the ratio of the difference between two quantities to the original quantity

5. Percent Change

6. Example 2 Finding Relative and Percent Change
The table below gives the average health insurance costs paid by employees of a company for years between 1998 and 2003.
Year
1998
1999
2000
2001
2002
2003
Cost ($)
627
677
740
768
972
1196
Find the relative and percent change in the health insurance paid by employees between 1998 and 2003.

7. Average Rate of Change
This measure uses the ratio of change in one quantity to the change in a second quantity.
An example of average rate of change is miles per hour, which is the ratio of change in distance (miles) to change in time (hours).

8. Average Rate of Change
Find the average rate of change in the health insurance paid by employees from 1998 to 2003 by finding the ratio of the change in costs to the change in years.
Year
1998
1999
2000
2001
2002
2003
Cost ($)
627
677
740
768
972
1196

9. Average Velocity

10. Example 3 Finding Average Velocity from a Table
The following table gives the height (in feet) above the ground of a ball that was tossed vertically upward at 32 feet per second from a three-story building
Time t (sec)
1.25
1.5
1.75 2
2.25
2.5
Height y
(ft) 45 42 37 30 21 10
a. Find the change in height between t = 1.25 and t = 2 seconds.
b. Find the relative change and percent change in height between t = 1.25 and
t = 2 seconds and interpret.
c. Find the average velocity (average rate of change) from t = 1.25 to t = 2
seconds and interpret.

11. Example 4 Finding Average Rate of Change from a Graph
The graph illustrates the height of the ball given in Example 3 against time. Find the average rate of change between the two points shown and interpret.

12. Average Rates of Change Symbolically
Symbolically, we have

13. Important Note:
Subtracting both sets of values in the reverse order will result in the same value.

14. Example 5 Finding Average Rate of Change using Function Notation
The height of the ball given in Example 3 is given by the rule
where f(t) is given in feet and t is given in seconds. Use this rule to find the average rate of change of the ball’s height between t = 0.5 second and t = 1 second and interpret the result.

15. Example 6 Interpreting Average Rates of Change
The following table gives the projected world population (in billions) for the given years. Create a line graph of the data. Find the average rate of change between 1950 and 2030 and interpret the result.
Year
1950
1960
1970
1980
1990
2000
2010
2020
2030
Population
in billions
2.7 3
3.8
4.5
5.2 6
6.9
7.5
8.1
A line graph is a scatterplot in which the points are connected with line segments.