# How do I find the average rate of change for a function between two given values?

Sep 9, 2014

Average rate of change is just another way of saying "slope".
For a given function, you can take the x-values and use them to calculate the y-values, then use the slope formula: $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

Example: Given the function f(x) = 3x - 8, find the average rate of change between 1 and 4.

f(1) = 3(1) - 8 = -5 and f(4) = 3(4) - 8 = 4

m = $\frac{4 - \left(- 5\right)}{4 - 1}$ = $\frac{9}{3}$ = 3 Surprised? No, because that is the slope between ANY two points on that line!

Example: f(x) = ${x}^{2} - 3 x$ , find the average rate of change between 0 and 2.

f(0) = 0 and f(2) = 4 - 6 = -2

m = $\frac{- 2 - 0}{2 - 0}$ = $\frac{- 2}{2}$ = -1
Since this function is a curve, the average rate of change between any two points will be different.

You would repeat the above procedure in order to find each different slope!

If you are interested in a more advanced look at "average rate of change" for curves and non linear functions, ask about the Difference Quotient.