Rate of change

The calculation

$$\frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$

The symbol $\Delta$ is a capital Greek delta meaning “change in”. The arithmetic is identical to the slope formula.

Slope or rate of change?

Same number, different question.

Term	Answers
slope	how steep is this line on the grid?
rate of change	how fast is this changing?

The second comes with units: $50$ becomes $50$ miles per hour. The units are always output per input, read straight off the axes, which is also the fastest check that you divided the right way round.

The sign

Positive means the output rises as the input rises; negative means it falls. A draining tank and a cooling drink both have negative rates, and the minus sign is carrying real information rather than signalling an error.

Why “average”

On a curve the steepness varies, so one number can only summarise an interval. On a straight line the rate is identical everywhere, so the word adds nothing there — see rate of change for the algebra showing why.