Average Rate of Change

Key Questions

How is average rate of change related to linear functions?

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

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

AJ Speller · · Sep 28 2014
How can the average rate of change be interpreted from a graph or a function?

Answer:

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.

SCooke · 1 · Oct 19 2015
How do I find the average rate of change for a function between two given values?

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: $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ $m=frac{y_2-y_1}{x_2-x_1}$

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

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

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

m = $\frac{- 2 - 0}{2 - 0}$ $frac{-2-0}{2-0}$ = $\frac{- 2}{2}$ $frac{-2}{2}$ = -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.

MrsRudolph221 · 1 · Sep 9 2014
What does average rate of change mean?

Answer:

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

Explanation:

Consider the following diagram:

when $x$ $x$ changes from $x 1$ $x1$ to $x 2$ $x2$ the value of the function changes from $y 1$ $y1$ to $y 2$ $y2$ . The average rate of change will be:
$\frac{y 2 - y 1}{x 2 - x 1}$ $(y2-y1)/(x2-x1)$ and it is, basically the slope of the blue line.

For example:
if $x 1 = 1$ $x1=1$ and $x 2 = 5$ $x2=5$
and:
$y 1 = 2$ $y1=2$ and $y 2 = 10$ $y2=10$
you get that:
Average rate of change $= \frac{10 - 2}{5 - 1} = \frac{8}{4} = 2$ $=(10-2)/(5-1)=8/4=2$

This means that for your function: $every time x increases of 1 then y increases of 2$ $color(red)("every time "x" increases of 1 then "y" increases of 2"$
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.

Gió · 2 · Jul 31 2015