Section 1.5

Functions and Change

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

Example 1 Finding Change

• Find the change in temperature if the temperature went from F to F. 79 93

Relative Change• Relative change is the ratio of the difference

between two quantities to the original quantity

Change Change from to Relative Change = = = Original Amount

A B B AA A

Percent Change

Percent Change Relative Change 100%

100%

=

= B AA

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.

Find the relative and percent change in the health insurance paid by employees between 1998 and 2003.

Year 1998 1999 2000 2001 2002 2003Cost ($)

627 677 740 768 972 1196

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).

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

Average Velocity

Change in DistanceChange in Time

Average Velocity

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 2.5Height 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.

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.

Average Rates of Change Symbolically

• Symbolically, we have

2 1

2 1

Change in OutputAverage Rate of Change =Change in Input

f x f xx x

Important Note:

• Subtracting both sets of values in the reverse order will result in the same value.

2 2

21 2

1 1

1

f x xx

x xxf

xfx

f

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.

216 32 30f t t t

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.