Chapter 8.6, Problem 35E

To determine

To explain why the average rate of change of a linear function is constant and the average rate of change of a quadratic or exponential function is not constant.

Explanation of Solution

Average rate of change of function is given by:

  $\text{Average rate of change}=\text{slope}=\frac{\text{change in }y}{\text{change in }x}=\frac{dy}{dx}$

Now,

Consider a standard linear function, $y=mx+c$

So,

  $\begin{array}{c}\text{Average rate of change}=\frac{dy}{dx}\\ =\frac{d\left(mx+c\right)}{dx}\\ =m\\ =\text{constant}\end{array}$

For a quadratic function in standard form, $y=a{x}^2+bx+c$

So,

  $\begin{array}{c}\text{Average rate of change}=\frac{dy}{dx}\\ =\frac{d\left(a{x}^2+bx+c\right)}{dx}\\ =2ax+b\end{array}$

  $\text{Since it contains variable }x\text{, so it is not constant}\text{.}$

Also,

For a exponential function in standard form, $y=a{e}^{bx}+c$

So,

  $\begin{array}{c}\text{Average rate of change}=\frac{dy}{dx}\\ =\frac{d\left(a{e}^{bx}+c\right)}{dx}\\ =ab{e}^{bx}\end{array}$

  $\text{Since it contains variable }x\text{, so it is not constant}\text{.}$

Thus,

The average rate of change of a linear function is constant and the average rate of change of a quadratic or exponential function is not constant.