AP® Precalculus · Unit 1 · Topic 1.2
Average Rate of Change
Average rate of change measures how much a function's output changes per unit of input over an interval, computed as (f(b) - f(a))/(b - a). Graphically, it is the slope of the secant line connecting two points on the curve.
On a road trip, your speedometer reading at any instant is almost never your average speed for the whole trip, because you sped up, slowed down, and maybe stopped for gas. Average rate of change ignores all that fluctuation and just compares where you started to where you ended. For a function f, the average rate of change over an interval from x = a to x = b is (f(b) - f(a))/(b - a): total change in output divided by total change in input.
Plot the two points (a, f(a)) and (b, f(b)) on the curve and draw a straight line through them. That line is a secant line, and its slope is exactly the average rate of change. This works because slope itself is rise over run, and here the rise is the change in f and the run is the change in x. The curve can wiggle, climb, or dip between a and b in any way it wants; the secant line only cares about the two endpoints.
This idea sets up everything else in the unit. As b gets closer to a, the secant line starts to hug the curve more tightly, which is the bridge toward instantaneous rate of change later on. For polynomials, average rate of change also hints at end behavior and curvature: a positive value means the function net increased over the interval, a negative value means it net decreased, and a value of zero just means the function landed back where it started, not that it stayed flat the whole time.
Key terms
- Average rate of change
- The ratio (f(b) - f(a))/(b - a), measuring net change in output per unit of input over an interval.
- Secant line
- A straight line connecting two points on a curve, whose slope equals the average rate of change between them.
- Interval
- The range of x-values, from a to b, over which the average rate of change is computed.
- Slope
- The ratio of vertical change to horizontal change between two points on a line.
Watch it explained
- How it worksHow does the average rate of change turn into the slope of a secant line on a curve?
- Worked exampleFind the average rate of change of f(x) = x² - 4x on the interval from x = 1 to x = 5.
- Worked exampleFind the average rate of change of f(x) = 2x from x = 0 to x = 3 and draw it as a secant slope.
- Common mistakeWhy doesn't an average rate of change of zero mean the function never moved?
Check yourself
End of topic.