Average Rate of Change Calculator
Find the average rate of change of a function f(x) on an interval [a, b] — the slope of the secant line — with f(a), f(b), Δy, Δx and the secant line equation.
How it works
Average rate of change on [a, b] = Δy/Δx = [f(b) − f(a)] / (b − a) = slope of the secant line through (a, f(a)) and (b, f(b))
The average rate of change measures how much a function’s output changes per unit of input over an interval. Geometrically it is the slope of the secant line joining the two endpoints of the graph. For a straight line it is the same on every interval; for curves it depends on the interval, and as b approaches a it approaches the derivative — the instantaneous rate of change.
Worked example
f(x) = x² + 3x on [1, 4]: f(1) = 1 + 3 = 4 and f(4) = 16 + 12 = 28, so Δy = 24 and Δx = 3. The average rate of change is 24/3 = 8, and the secant line is y = 8x − 4.
Sources: OpenStax College Algebra 2e — 3.3 Rates of Change and Behavior of Graphs · OpenStax Calculus Volume 1 — 3.1 Defining the Derivative
Frequently asked questions
How do you find the average rate of change?
Evaluate the function at both ends of the interval, subtract the outputs, and divide by the difference of the inputs: [f(b) − f(a)] / (b − a).
Is average rate of change the same as slope?
Yes — it is the slope of the secant line between the two points. For a linear function it equals the slope of the line itself.
What is the difference between average and instantaneous rate of change?
The average uses two points; the instantaneous rate is the limit as the interval shrinks to one point, which is the derivative f′(a).
What does a negative average rate of change mean?
That the function ended lower than it started on that interval, i.e. it decreased on average (it may still have gone up in between).
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