Copyright © Cengage Learning. All rights reserved. 2 Functions Copyright © Cengage Learning. All rights reserved.

Average Rate of Change and Revisiting Increasing/Decreasing Functions 2.5 Copyright © Cengage Learning. All rights reserved.

Slope and Rate of Change

Slope and Rate of Change Note: If f(x)=ax+b then the average rate of change = slope. The average rate of change for a linear function is constant or the same between any two points, therefore we usually just call it rate of change for linear functions instead of average rate of change. But the approach is the same regardless of they type of function

Example 3 – Slope and Rate of Change A dam is built on a river to create a reservoir. The water level f (t) in the reservoir at time t is given by f (t) = 4.5t + 28 where t is the number of years since the dam was constructed and f (t) is measured in feet. (a) Sketch a graph of f. (b) What is the slope of the graph? (c) At what rate is the water level in the reservoir changing?

Example 3 – Solution (a) A graph of f is shown in Figure 2. Water level as a function of time Figure 2

Example 3 – Solution cont’d (b) The graph is a line with slope 4.5, the coefficient of t. (c) The rate of change of f is 4.5, the coefficient of t. Since time t is measured in years and the water level f (t) is measured in feet, the water level in the reservoir is changing at the rate of 4.5 ft per year. Since this rate of change is positive, the water level is rising.

Increasing and Decreasing Functions The function f is said to be increasing when its graph rises and decreasing when its graph falls. We have the following definition.

There is a difference between locating the highest and lowest points on a graph in a region around an open interval (locally) and locating the highest and lowest points on the graph for the entire domain. The y-coordinates (output) at the highest and lowest points are called the absolute maximum and absolute minimum, respectively