# Average Rate of Change

## Key Questions

• The average rate of change is constant for a linear function.

Another way to state this is that the average rate of change remains the same for the entire domain of a linear function.

If the linear function is $y = 7 x + 12$ then the average rate of change is 7 over any interval selected.

Slope intercept form
$y = m x + b$, where $m$ is the slope.

The rate of change is the slope of the graph.

#### Explanation:

It really doesn't make much sense to try to apply this to nonlinear functions, and you certainly cannot apply an "average" value to a non-linear function unless you first linearize it. Even then, the interpretation of what that "average" means must be carefully understood.

• 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.

The average rate of change of a function $y = f \left(x\right)$, for example, tells you of how much the value of the function changes when $x$ changes.

#### Explanation:

Consider the following diagram:

when $x$ changes from $x 1$ to $x 2$ the value of the function changes from $y 1$ to $y 2$. The average rate of change will be:
$\frac{y 2 - y 1}{x 2 - x 1}$ and it is, basically the slope of the blue line.

For example:
if $x 1 = 1$ and $x 2 = 5$
and:
$y 1 = 2$ and $y 2 = 10$
you get that:
Average rate of change$= \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2$

This means that for your function: color(red)("every time "x" increases of 1 then "y" increases of 2"
Obviously your function is not a perfect straight line and it will change differently inside that interval but the average rate can only evaluate the change between the two given points not at each individual point.