Average Rate of Change Calculator

What is the average rate of change?

The average rate of change measures how much a function’s output value changes, on average, for each unit of change in its input over a chosen interval. Given two points on the graph of a function, $(x_1, y_1)$ and $(x_2, y_2)$, it tells you the slope of the straight line (the secant line) connecting them. A positive result means the function rises across the interval, a negative result means it falls, and zero means the endpoints sit at the same height.

Formula

The denominator $x_2 - x_1$ must not be zero. When the two x-values are equal there is no horizontal interval to average over, so the rate of change is undefined.

How to use

1. Enter the coordinates of the first point: $x_1$ and $y_1$.
2. Enter the coordinates of the second point: $x_2$ and $y_2$.
3. The calculator divides the change in y by the change in x and shows the average rate of change automatically.
4. If $x_1$ and $x_2$ are equal, the result stays empty because the value would require dividing by zero.

Worked example

Take the points $(1, 2)$ and $(4, 11)$:

So the function changes by an average of 3 units of y for every 1 unit increase in x across this interval.

FAQ

How is the average rate of change related to slope? For a straight line, the average rate of change between any two points equals the line’s slope. For a curve, it equals the slope of the secant line joining the two chosen points. You can explore that relationship further with the slope calculator.

Why is the result empty when x₁ equals x₂? The formula divides by $x_2 - x_1$. If the two x-values are identical, that denominator is zero and division is undefined, so the calculator returns no value. To express a change as a proportion instead, try the percentage calculator.