Average Rate of Change

Pre-Calculus · Axiom Academy

How fast does a function change between two points? It's the slope of the line that joins them. 1. The Secant Slope: Rise Over Run Take the curve f(x) = x^2 and mark two points on it: A = (1, 1) and B = (3, 9) . Draw the line through them — the secant line . Its slope is the vertical change (the rise ) divided by the horizontal change (the run ), and that single number is the average rate of change of f from 1 to 3 . 2. The Formula, and What Its Sign Tells You Replace the specific numbers with a and b and you have the general definition. The same rise-over-run, written for any function and any two inputs: The sign of that number tells the story at a glance — read the secant line as you slide b around: The secant tilts up — output is increasing on average. For x^2 on [1,3] the rate is +4 . The secant tilts down — output is decreasing on average. For x^2 on [-3,-1] the rate is -4 . The secant is flat — equal heights at the ends, no net change. For x^2 on [-2,2] the rate is 0 . For a straight line like 2x + 1 the rate is the same on every interval — always 2 . Because , the average rate of change of x^2 on [a,b] is just a + b . Check it: on [1,3] that's 1 + 3 = 4 , exactly the slope we found. 3. Closing the Gap: Secant Becomes Tangent

This is the written version of the interactive lesson above. See the full Pre-Calculus course.