Average Rate of Change Calculator
Find the average rate of change of a function over an interval [a, b] — the slope of the secant line connecting the two endpoints.
Function f(x)
Slope of the secant line between a and b
- 1
Evaluate f(a)
f(1) = 1 - 2
Evaluate f(b)
f(4) = 16 - 3
Change in output
f(b) − f(a) = 16 − 1 = 15 - 4
Change in input
b − a = 4 − 1 = 3 - 5
Average rate of change
(15) ÷ (3) = 5Slope of the secant line through (a, f(a)) and (b, f(b)).
How does this calculator work?
The average rate of change of f over [a, b] is (f(b) − f(a)) / (b − a), the slope of the secant line joining the endpoints. Pick a function and two x-values; the tool returns this slope plus f(a) and f(b), and plots the curve with its secant line.
Formula
How this is calculated
Select a function f(x) and two interval endpoints, a and b. The calculator evaluates the function at both endpoints to get f(a) and f(b), then divides the change in output by the change in input.
The average rate of change equals (f(b) − f(a)) / (b − a). Geometrically this is the slope of the secant line through the points (a, f(a)) and (b, f(b)) on the graph. It tells you, on average, how fast the function rises or falls per unit of x across the interval — distinct from the instantaneous rate (the derivative) at a single point.
The endpoints a and b must be different, otherwise the denominator b − a is zero and the slope is undefined. The chosen function must also be defined at both endpoints (for example, √x needs x ≥ 0 and 1/x needs x ≠ 0); if an endpoint falls outside the domain the result is reported as invalid. The plot shows the curve with the secant line drawn between the two endpoints.
Frequently asked questions
The average rate of change is the slope of the secant line over an entire interval [a, b], while the derivative is the instantaneous rate of change (slope of the tangent line) at a single point. As b approaches a, the average rate of change approaches the derivative at a.
The formula divides by (b − a). If a equals b, that denominator is zero and the slope is undefined, so the calculator requires two distinct endpoints.
Yes. A negative value means the function decreases overall across the interval — f(b) is less than f(a). A value of zero means f(a) and f(b) are equal even if the function rose and fell in between.
Also known as
TG we-Calculate Editorial Team. (2026). Average Rate of Change Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/average-rate-of-change-calculator
TG we-Calculate Editorial Team. "Average Rate of Change Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/average-rate-of-change-calculator.
TG we-Calculate Editorial Team, "Average Rate of Change Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/average-rate-of-change-calculator
@misc{wecalculate_average_rate_of_change_calculator, title = {Average Rate of Change Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/average-rate-of-change-calculator}}, year = {2026}, note = {TG we-Calculate} }
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