Average rate of change describes how much an output changes, on average, for each unit change in an input over an interval. It is the slope of the line joining two points, even when the original relationship is curved.
Use change in output over change in input
For points (x1, y1) and (x2, y2), average rate of change is (y2 - y1) / (x2 - x1). The numerator measures the output change and the denominator measures the input change.
A result of 2 means the output increased by an average of two units for each one unit of input across the interval. A negative result means the output decreased as the input increased.
Average rate and slope
The same calculation is commonly called the slope of a secant line. On a graph, connect the two points and read how far the line rises or falls for each horizontal unit.
For a straight-line function, the average rate is the same on every interval. For a curved function, it can change from one interval to another and does not describe every point between the endpoints.
Do not confuse it with percentage change
Percentage change divides the difference by the original value and multiplies by 100. Average rate of change divides by the change in the input. A price may rise by $10 per item and also rise by 20%, but those are different questions.
Use the Percentage Change Calculator for a before-and-after percentage comparison. Use the Rate of Change Calculator when the input and output units, such as time and distance, are part of the relationship.
Check the interval before calculating
The input values must be different. If x2 equals x1, the line is vertical and the average rate is undefined because division by zero is not valid.
Keep the units attached to the result. If distance changes from 10 to 18 kilometers while time changes from 2 to 4 hours, the average rate is 4 kilometers per hour, not simply the number 4.
A worked example
Worked example: if a distance rises from 10 to 18 kilometers while time rises from 2 to 4 hours, the average rate is (18 - 10) / (4 - 2) = 4 kilometers per hour.
Keep the input and output units attached to the answer and name the interval. That makes it clear whether the result is a rate, a slope, or a percentage comparison.
Average rate of change is a compact slope calculation, but its meaning depends on the interval, the units, and whether the question asks for a rate or a percentage.
COMMON QUESTIONS
Frequently asked questions
What is the formula for average rate of change?
Subtract the first output from the second output, then divide by the second input minus the first input: (y2 - y1) / (x2 - x1).
Is average rate of change the same as slope?
It is the slope of the secant line through two points. For a straight line it matches the constant slope; for a curve it describes only the selected interval.
What happens when the input change is zero?
The rate is undefined because division by zero is not valid. The two points form a vertical line rather than a finite rate.