Average Rate of Change Calculator

The average rate of change measures how much a function output changes, on average, for each unit of change in its input over an interval. Given two points (x1, y1) and (x2, y2), subtract the initial y value from the final y value, then divide by the final x value minus the initial x value. The formula is (y2 - y1) ÷ (x2 - x1). Geometrically, this value is the slope of the secant line connecting the two points. A positive result means the function increased overall across the interval, while a negative result means it decreased. A result of zero means the endpoints have equal outputs, although the function may still rise or fall between them. The units are output units per input unit. For example, if y measures distance in kilometers and x measures hours, the rate is kilometers per hour. The initial and final x values cannot be equal because division by zero is undefined. Reversing both points does not change the answer: both the numerator and denominator change sign, so their quotient remains the same. The calculation uses only the endpoints and does not describe every instantaneous change inside the interval. For a curved function, instantaneous rate of change is found with a derivative rather than a secant slope. This concept appears in algebra, calculus, physics, economics, and data analysis. It can describe average velocity, cost per added item, temperature change per hour, or population growth per year. Enter coordinates directly from a graph, table, or function evaluation. The calculator shows the substituted formula as well as the result, making signs and subtraction easy to inspect. Keep the original measurement units when interpreting the answer, and remember that an endpoint average can hide variation occurring between the two selected points. Choosing meaningful endpoints is therefore essential for a useful interpretation.