About this tool
The Average Rate of Change & Secant Line Calculator is a free, in-browser tool that measures how fast a function changes on average across an interval. Given f(x) and two endpoints a and b, the average rate of change is the difference quotient m = (f(b) − f(a)) / (b − a) — the slope of the straight secant line joining the two points (a, f(a)) and (b, f(b)) on the curve. This is the discrete, interval version of the derivative, which is the instantaneous rate at a single point.
All of the arithmetic runs locally in your browser using a small expression parser, so nothing you enter leaves your device. Type the function with x as the variable (for example x^2, sin(x), or 1/x) and the two endpoints, and the calculator returns the average rate of change (the secant slope), both endpoints (a, f(a)) and (b, f(b)), and the secant line itself in slope-intercept form y = m·x + c and point-slope form.
The secant line is the foundation of differential calculus: as the second point b approaches a, the secant slope approaches the tangent slope, and the average rate of change becomes the instantaneous rate — the derivative f′(a). For f(x) = x² on [1, 3] the average rate of change is (9 − 1)/(3 − 1) = 4, so the secant line is y = 4x − 3.