what is the only type of function that has a constant average rate of change?\nchoose the correct answer…

Brief Explanations:

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