Average Rate of Change

What is the Average Rate of Change? You might guess that the average rate of change of a function is how quickly the function changes. You’d be right! It’s the speed, on average, that a function increases or decreases between two points. Basically, the average rate of change tells you how much f(x) increases for each increase of x between two points.

Determining the Average Rate of Change The nice thing about the average rate of change is that it doesn’t matter how you get from one point to another – it just matters where you start and where you end up. To calculate the average rate of change of a function between two points, just calculate how much the function changes between those two points, then divide by how much the independent variable changes. The equation for the average rate of change is rate = ∆f/∆x. These two functions have the same average rate of change between (-5, -6) and (5,6).

Example What is the average rate of change of the function f(x) = 3x2 – 5x + 2 between x = 0 and x = 3?

Solution When x = 0, f(x) = 2. When x = 3, f(x) = 14. Remember, average rate of change = ∆f/∆x ∆x = 3, because x increases from 0 to 3. ∆f = 12, because f(x) increases from 2 to 14. Thus, the average rate of change is 12/3 = 4.

Graph Note that the average rate of change of the function between two points is equal to the slope of the line that connects the two points.