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What Is the Average Rate of Change, and How Do You Find It?

average rate of change: girl doing her homework

Average rate of change is the rate at which one value within a function changes in relationship to another. The average rate of change is usually used to determine the slope of a graphed function.

Let's explore how to find the average rate of change and use linear and nonlinear functions to demonstrate it.

How to Find the Average Rate of Change

The average rate of change of a function corresponds to the slope of the line, which connects two endpoints of a given interval (known as the secant line). Here is the average rate of change formula:

average rate of change formula

The average rate of change of the function has the change in y-values in the numerator and the change in x-values as the denominator. We’ll subtract the x and y-values of the second point from the x and y-values of the first point, then simplify.

Linear Rate of Change

Since linear functions are always straight lines, their average rate of change will remain constant no matter what set of points you plug into the slope formula.

Let’s find the average rate of change for the linear function below:

slope formula

linear rate of change

Image credit: Desmos

average rate of change for linear

Nonlinear Rate of Change

A nonlinear function, however, doesn’t have a constant rate of change. It will have a different slope depending on what points you use in the average rate of change formula.

So, you can find the rate of change by forming a straight secant line segment that goes through two point, which we’ll use in the average rate of change formula:

nonlinear rate of change

Image credit: Desmos

average rate of change for nonlinear

When to Use the Average Rate of Change Formula

The average rate of change formula is used to find the slope of a graphed function. To find the average rate of change, divide the change in y-values by the change in x-values.

Finding the average rate of change is particularly useful for determining changes in measurable values like average speed or average velocity.

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