# Average rate of change review

CCSS Math: HSF.IF.B.6
Review average rate of change and how to apply it to solve problems.

## What is average rate of change?

The average rate of change of function $f$ over the interval $a\leq x\leq b$ is given by this expression:
$\dfrac{f(b)-f(a)}{b-a}$
It is a measure of how much the function changed per unit, on average, over that interval.
It is derived from the slope of the straight line connecting the interval's endpoints on the function's graph.

## Finding average rate of change

### Example 1: Average rate of change from graph

Let's find the average rate of change of $f$ over the interval $0\leq x\leq 9$:
We can see from the graph that $f(0)=-7$ and $f(9)=3$.
\begin{aligned} \text{Average rate of change}&=\dfrac{f(9)-f(0)}{9-0} \\\\ &=\dfrac{3-(-7)}{9} \\\\ &=\dfrac{10}{9} \end{aligned}

### Example 2: Average rate of change from equation

Let's find the rate of change of $g(x)= x^3 - 9x$ over the interval $1\leq x\leq 6$.
$g(1)=1^3-9\cdot1=-8$
$g(6)=6^3-9\cdot 6=162$
\begin{aligned} \text{Average rate of change}&=\dfrac{g(6)-g(1)}{6-1} \\\\ &=\dfrac{162-(-8)}{5} \\\\ &=34 \end{aligned}
Problem 1
What is the average rate of change of $g$ over the interval $-8\leq x\leq -2$?

Want to try more problems like this? Check out this exercise.