How fast it changed, on average

The slope between two points on a curve, what it measures, and what its units are.

A car goes 120 kilometres in two hours. Nobody claims it held 60 kilometres per hour the whole way; the number is the average, and it is worked out from the two ends of the journey alone. Functions get the same treatment.

Take a rule, pick two inputs, and divide the change in output by the change in input. Run it over several intervals of the same function.

For $x^2$ the four answers are 1, 3, 5 and 7. The function is getting steeper as you move right, and a single number cannot describe the whole curve. For $3x + 5$ every interval gives 3, because a straight line has the same steepness everywhere.

That quantity has a name and a formula. The average rate of change of $f$ over the interval from $a$ to $b$ is the change in output divided by the change in input.

Geometrically this is the slope of the straight line drawn through the two points $(a, f(a))$ and $(b, f(b))$. That line is called a secant line, from the Latin for cutting, because it cuts the curve at two places.

It is the same formula you already use for the slope of a line, $\frac{y_2 - y_1}{x_2 - x_1}$, with the two points taken off the curve. That is why a linear function gives the same answer over every interval: there the secant line and the function are the same line.

The order of the two endpoints does not matter, as long as you keep it consistent. Swapping $a$ and $b$ flips the sign of the top and of the bottom at once, and the two flips cancel. Swapping only one of them is an error, and it produces an answer with the wrong sign.

One worked example in full. Find the average rate of change of $f(x) = x^2 - 4x$ from $x = 1$ to $x = 5$.

1. Work out the output at each end: $f(1) = 1 - 4 = -3$.
2. And the other: $f(5) = 25 - 20 = 5$.
3. Change in output: $5 - (-3) = 8$.
4. Change in input: $5 - 1 = 4$.
5. Divide: $\frac{8}{4} = 2$.

So on average the output rose 2 units for every 1 unit of input across that stretch. Note that the function actually falls at first and rises later; the average hides that entirely, which is exactly what an average does.

Units, and what the number means

In a modelling question the answer needs units, and they come straight from the formula: the units of the output divided by the units of the input. If $P(t)$ is a population in people and $t$ is in years, the average rate of change is in people per year.

The three weeks give 1, about 1.43 and about 0.43 centimetres per day. A negative answer would mean the quantity fell over that stretch, and zero would mean it ended where it started, whatever it did in between.