Answer
$(a)$ $\frac{f(x_2)-f(x_1)}{x_2-x_1}$
$(b)$ Simply inputing the points and simplifying will show us that the average rate of change equals to the slope.
See the calculations below.
Work Step by Step
$(a)$ We simply input $x_1$ and $x_2$ into the equation of the average rate of change (same as the slope) and we will get the following equation:
Average rate of change $=\frac{f(x_2)-f(x_1)}{x_2-x_1}$
$(b)$ We cans imply input the points and simplify:
$\frac{f(x_2)-f(x_1)}{x_2-x_1}=\frac{(ax_2+b)-(ax_1+b)}{x_2-x_1}=\frac{ax_2+b-ax_1-b}{x_2-x_1}=\frac{ax_2-ax_1}{x_2-x_1}=\frac{a(x_2-x_1)}{x_2-x_1}=a$
As we can see, the average rate of change is the same as the slope $a$.
