Average Rate of Change

A function's output changes as its input changes. The average rate of change measures how fast: the change in output divided by the change in input. Take $f\left(x\right) = {x}^{2}+3x$ from x = 0 to x = 2. The output goes from $f\left(0\right) = 0$ to $f\left(2\right) = 10$, a change of 10, while the input changes by 2. So the rate is 10 over 2, which is 5: on average the output grows 5 for every 1 the input grows.

In general, from x = a to x = b the average rate of change is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$. On the graph, (a, f(a)) and (b, f(b)) are two points, and this fraction is the slope of the straight line through them: rise over run.