"We averaged 80 km/h." You've said something like it after every long car journey, and you computed it without a moment's thought: the odometer read 18{,}240 km when you left and 18{,}480 km when you arrived, the clock said the trip took 3 hours, so

Total change divided by time taken. That's it — that's the whole idea of this page, and you already own it. But look closer at what the number hides. On that trip you sat in a roadworks jam for twenty minutes, completely stationary — speed 0. Later, on the empty motorway, the speedometer touched 120. The "80" you reported is neither of those; it never appeared on the speedometer for more than an instant. It is a single number summarising the whole messy, variable journey: if you had somehow driven at one perfectly steady speed and still covered 240 km in 3 hours, that speed would have been 80 km/h.

That is what an average rate of change is, for any changing quantity — a growing population, a cooling coffee, a rising share price, a draining tank. Take the total change in the thing you care about, divide by the change in the input (usually time), and you get the steady rate that would have produced the same overall change.

The definition

For a function f over an interval from x = a to x = b, the average rate of change is the change in the output divided by the change in the input:

The capital Greek delta, \Delta, is mathematics' shorthand for "change in": \Delta x = b - a is how far the input moved, and \Delta f = f(b) - f(a) is how far the output moved in response. Read the fraction aloud and it says exactly what the road trip said: output change per unit of input change.

And that fraction should look deeply familiar — it is exactly the slope of a line, rise over run, just written with function values instead of coordinates. This is not a coincidence; it's the geometric heart of the whole idea, and it's where we go next.

Two things to keep straight from the start. First, the average rate carries units: output units per input unit — km per hour, people per year, degrees per minute. Second, it has a sign: if f ends lower than it started, \Delta f is negative and so is the rate — a draining tank has a negative rate of change in litres per minute.

It is the slope of the secant line

Now draw the picture. Mark the two points \big(a, f(a)\big) and \big(b, f(b)\big) on the graph of f and join them with a straight line. That line — a line that cuts the curve at two points — is called a secant line (from the Latin secare, "to cut"), and the average rate of change is its slope: rise f(b) - f(a) over run b - a.

Try it live on the parabola f(x) = x^2. Drag the sliders to move the two endpoints and watch the bold secant line tilt as the interval changes. Three experiments worth running: put a = 1, b = 3 and note the steepness; slide both points to the left half of the parabola and watch the slope go negative (the function is falling there); and — the fun one — set a = -2, b = 2. The secant goes perfectly flat: the parabola climbed and fell by exactly the same amount, so its average rate of change over that interval is zero, even though it was never standing still except for one instant at the bottom.

That flat secant is the graph-world version of the roadworks jam: an average can be tame while the function underneath is doing wild things. The secant only ever consults the two endpoints.

Worked example: a quadratic, step by step

Take f(x) = x^2 on the interval from x = 1 to x = 3, and follow the recipe deliberately.

Step 1 — evaluate at both endpoints. f(1) = 1^2 = 1 and f(3) = 3^2 = 9.

Step 2 — form the two changes. The rise is \Delta f = f(3) - f(1) = 9 - 1 = 8; the run is \Delta x = 3 - 1 = 2.

Step 3 — divide.

Over this stretch the function rises, on average, 4 units of output for every 1 unit of input. Watch the same computation happen geometrically — the secant, its run, its rise:

One more, with a negative answer. Same parabola, but from x = -3 to x = 1: f(-3) = 9 and f(1) = 1, so

The function fell overall, so the average rate is negative — even though, near the right-hand end, the parabola had already turned around and was climbing again. Order matters inside each difference (top and bottom must both run a \to b), but it doesn't matter which point you call a: swap them and both differences flip sign, so the fraction survives untouched.

No formula? No problem: rates straight from data

Here is something quietly powerful about \tfrac{f(b) - f(a)}{b - a}: it never asks what f is. It only needs two values. So it works just as well on measured data with no formula in sight — which is how it earns its living in geography, biology, economics and medicine. Suppose a census records a town's population:

Average rate of change from 2000 to 2010:

and from 2010 to 2020:

Both rates are positive — the town grew in both decades — but comparing them tells a story a single number can't: growth slowed to a third of its earlier pace. Town planners, epidemiologists tracking case counts, and analysts reading revenue tables all live on exactly this comparison: compute the average rate on successive intervals and watch how it changes. Notice the units did their job too — "people per year" fell straight out of dividing a change in people by a change in years.

Shrinking the interval — a peek over the horizon

A question should be nagging at you. The average over a wide interval blurs out detail (the traffic jam vanished into the "80"). So what if we made the interval narrower? Wouldn't the average then describe what's happening near a point more and more faithfully?

Let's try it, on f(x) = x^2, anchoring the left end at x = 1 and pulling the right end in:

The averages march in single file: 4,\ 3,\ 2.5,\ 2.1,\ 2.01,\ \dots — homing in on 2. There's a slick reason. For f(x) = x^2 the average rate between any two points has a beautiful closed form:

The average rate of x^2 from a to b is simply a + b! (Check the table: 1 + 3 = 4, 1 + 2 = 3, 1 + 1.5 = 2.5…) So as b slides toward 1, the average slides toward 1 + 1 = 2, and nothing can stop it.

Pause on what just happened, because it is the doorway to the rest of calculus. The averages aren't just wandering — they are converging, as if the function has a definite steepness at the single point x = 1, and the shrinking averages are sniffing it out. That limiting value — the "instantaneous" rate of change, the speedometer reading rather than the trip average — is the derivative, and chasing it down properly is exactly where this thread of pages is headed, starting with a tidier notation for the shrinking interval: the difference quotient. For now, hold on to the picture: average rate over an interval first; squeeze the interval later.

Watch it explained

Sal Khan introduces the average rate of change of a function over an interval, and shows how it connects to slope — a good second pass over everything above.