The Gradient of a Line

The gradient of a straight line measures its steepness — how much it rises for each step across. Between two points $(x_1, y_1)$ and $(x_2, y_2)$:

The sign tells the direction:

• Positive gradient — the line goes up from left to right (uphill).
• Negative gradient — the line goes down from left to right (downhill).
• Zero gradient — a flat, horizontal line.
• Undefined gradient — a vertical line (the run is 0, and we cannot divide by 0).

Worked example. The gradient through $(1, 2)$ and $(3, 8)$ is $m = \dfrac{8 - 2}{3 - 1} = \dfrac{6}{2} = 3$: the line rises 3 units for every 1 across.