How to Calculate and Interpret Average Rate of Change in Math

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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 (x1,y1) and (x2,y2), it is sufficient that you find the slope of the line passing through the points.

Theory

Average Rate of Change

Given two points (x1,y1) = (x1,f(x1)) and (x2,y2) = (x2,f(x2)), you have

average rate of change = change in y change in x = y2 y1 x2 x1 = f(x2) f(x1) x2 x1

average rate of change = change in y change in x = y2 y1 x2 x1 = f(x2) f(x1) x2 x1

The formula says that you can choose the version of the three fractions you need.

The figure below shows what this looks like graphically.

Average rate of change of a function

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 (x1,y1), where y1 = f(x1), and B has the coordinate (x2,y2), where y2 = f(x2). The red arrow between x1 and x2 indicates the change in the x-direction, and the red arrow between f(x1) and f(x2) 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 x1 = 12, x2 = 17, and that the change in distance traveled is

220mi 0mi = 220mi

The change in duration is 17h 12h = 5h. You enter this into the formula for average rate of change to find the average speed v:

v = 220mi 5h = 44mph.

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