Slope measures change · Math Foundry

The idea

Slope measures how much the output changes per unit of input. A line with slope 2 rises 2 units whenever $x$ moves right by 1 — this "change in $y$ per change in $x$ " is exactly the idea that becomes the derivative in calculus.

Real-world: filling a water tank

A tank begins with 10 litres and a pump adds 3 litres every minute, so on a volume-versus-time graph the slope is 3 litres per minute. After 4 minutes the volume is $V = 10 + 3(4) = 22$ litres — the slope is exactly the amount added each minute.

For y=2x+1, every +1 in x produces +2 in y.

Worked example

For points $(1,2)$ and $(4,8)$ :

\begin{aligned} \Delta y &= 8-2 = 6, \\ \Delta x &= 4-1 = 3, \\ \text{slope} &= \frac{\Delta y}{\Delta x} = \frac{6}{3} = 2. \end{aligned}

Cheat sheet

$\text{slope} = \dfrac{\Delta y}{\Delta x}$

Slope is "rise over run": how much $y$ changes for each unit increase in $x$ .
A horizontal line has slope $0$ ; a vertical line has an undefined slope.
Positive slope rises left to right; negative slope falls left to right.
Slope is a constant rate of change — the same idea generalizes to the derivative for curves.