# Average Rate of Change | Mathematics

August 24, 2015

Miles per gallon or miles per hour are calculations you’re probably familiar with. Did you know they’re actually average rate of change calculations? Rate of change describes the average rate one quantity is changing compared to another quantity changing. For the TASC Test Assessing Secondary Completion™ Mathematics subtest, you must be able to find the average rate of change based on a function’s graph or table.

Find the Average Rate of Change From a Graph

Let’s use the example of miles per hour. If you went on a road trip and recorded the distance you traveled every few minutes, you could say the distance s traveled is a function of the time that’s passed t. This is also written as:

s(t) = total distance traveled at time t

NYU's Courant Institute of Mathematical Sciences offers a graphed representation of a function s(t):

To find the average rate of change in the speed between two points, you must divide the distance traveled by the time elapsed. Let’s look at the average speed between 1:00 and 4:00.

• To find the time elapsed, calculate 4 minus1 to get 3 hours.
• To find the distance traveled, subtract the distance at 1:00 from the distance at 4:00:     200 – 50 to get 150 miles.
• Finally, to find the average speed, divide 150 miles by 3 hours to get 50 miles per hour. This is your average rate of change.

It’s important to note that your average rate of change (average speed) will differ throughout your trip. For example, between 2:00 and 3:00:

• The time elapsed is 1 hour (3 minus 2).
• The distance traveled is 65 miles (by subtracting the distance at 2:00 from the distance at 3:00: 140 – 75).
• The average speed (average rate of change) is 65 miles per hour (by dividing 65 miles by 1 hour).

Find the Average Rate of Change from a Table

The average rate of change can also be found given a table of values for a function.

What is the average rate of change for y(x) over the interval -5<x<-2?

When x is equal to -5, y(x) is equal to 6.By referencing the table:

• When x is equal to -2, y(x) is equal to 0.

To find the average rate of change of y(x) with respect to x, we must find the change in y divided by the change in x: (“change” is represented by the delta symbol: Δ)

Δy / Δx   à   (0) – (6) / (-2) – (-5)   à   (-6) / (3) = -2

Continue your practice of calculating the average rate of change for the TASC test at Khan Academy, and take our rate-of-change quiz by clicking HERE!

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