Question

what is the only type of function that has a constant average rate of change?
choose the correct answer below.
a. quadratic function
b. step function
c. absolute value function
d. linear function

Explanation:

The average rate of change of a function \(y = f(x)\) over the interval \([x_1,x_2]\) is given by \(\frac{f(x_2)-f(x_1)}{x_2 - x_1}\).
For a linear function \(y=mx + b\), \(\frac{(m x_2 + b)-(m x_1 + b)}{x_2 - x_1}=\frac{m(x_2 - x_1)}{x_2 - x_1}=m\) (a constant).
A quadratic function \(y = ax^{2}+bx + c\) has an average rate of change \(\frac{(a x_2^{2}+bx_2 + c)-(a x_1^{2}+bx_1 + c)}{x_2 - x_1}=\frac{a(x_2^{2}-x_1^{2})+b(x_2 - x_1)}{x_2 - x_1}=a(x_1 + x_2)+b\) (not constant).
A step - function has discontinuous jumps and its average rate of change is not constant over all intervals.
An absolute - value function \(y=\vert x\vert\) (for example, \(y = \vert x\vert=

$$\begin{cases}x, &x\geq0\\-x, &x < 0\end{cases}$$

\)) has different slopes (\(1\) for \(x>0\) and \(- 1\) for \(x < 0\)) in different intervals, so its average rate of change is not constant over all intervals.

Answer:

D. linear function