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)

Δ*y* / Δ*x *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**!