The average rate of change is the rate of change over an interval. That is, you should find the rate of change between two values on the first axis. To find the average rate of change between the points $({x}_{1},{y}_{1})$ and $({x}_{2},{y}_{2})$, it is sufficient that you find the slope of the line passing through the points.

Theory

Given two points $({x}_{1},{y}_{1})=({x}_{1},f({x}_{1}))$ and $({x}_{2},{y}_{2})=({x}_{2},f({x}_{2}))$, you have

$$\begin{array}{llll}\hfill \text{averagerateofchange}& =\frac{\text{changein}y\text{}}{\text{changein}x\text{}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{f({x}_{2})-f({x}_{1})}{{x}_{2}-{x}_{1}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill \end{array}$$

$$\begin{array}{llll}\hfill \text{averagerateofchange}& =\frac{\text{changein}y\text{}}{\text{changein}x\text{}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =\frac{f({x}_{2})-f({x}_{1})}{{x}_{2}-{x}_{1}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill \end{array}$$

The formula says that you can choose the version of the three fractions you need. The figure below shows what this looks like graphically.

In the figure, you see the function $f(x)$ (blue graph). You draw the straight line (red line) between $A$ and $B$. The point $A$ has the coordinate $({x}_{1},{y}_{1})$, where ${y}_{1}=f({x}_{1})$, and $B$ has the coordinate $({x}_{2},{y}_{2})$, where ${y}_{2}=f({x}_{2})$. The red arrow between ${x}_{1}$ and ${x}_{2}$ indicates the change in the $x$-direction, and the red arrow between $f({x}_{1})$ and $f({x}_{2})$ indicates the change in the $y$-direction. Study the drawing carefully and make sure you understand it!

Example 1

**You’re going on a trip to a friend’s cabin. You start the trip at 12:00 and arrive at 17:00. It is 220 miles to the cottage. What was the average speed of the car? **

You see that ${x}_{1}=12$, ${x}_{2}=17$, and that the change in distance traveled is

$$220\phantom{\rule{0.17em}{0ex}}\text{mi}-0\phantom{\rule{0.17em}{0ex}}\text{mi}=220\phantom{\rule{0.17em}{0ex}}\text{mi}$$ |

The change in duration is $17\phantom{\rule{0.17em}{0ex}}\text{h}-12\phantom{\rule{0.17em}{0ex}}\text{h}=5\phantom{\rule{0.17em}{0ex}}\text{h}$. You enter this into the formula for average rate of change to find the average speed $\stackrel{}{v}$:

$$\stackrel{}{v}=\frac{220\phantom{\rule{0.17em}{0ex}}\text{mi}}{5\phantom{\rule{0.17em}{0ex}}\text{h}}=44\phantom{\rule{0.17em}{0ex}}\text{mph}.$$ |