The gradient of a line

What gradient measures

• The gradient of a line is its steepness — how much it rises or falls for each step across.
• It is the same all the way along a straight line: pick any two points on it and you get the same value.
• Gradient is written as $m$.

The gradient formula

$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$

• rise ($y_2 - y_1$) — the change in $y$ (up is positive, down is negative).
• run ($x_2 - x_1$) — the change in $x$ (right is positive).
• Subtract the $y$-values on top and the $x$-values on the bottom, in the same order.

Reading the sign and size of a gradient

• Positive gradient — the line goes up from left to right (rise and run have the same sign).
• Negative gradient — the line goes down from left to right.
• Zero gradient — a horizontal line ($y_2 - y_1 = 0$); it never rises.
• Undefined gradient — a vertical line ($x_2 - x_1 = 0$); you would divide by zero, so it has no gradient.
• The bigger the size of $m$, the steeper the line: a gradient of $3$ is steeper than a gradient of $\tfrac{1}{2}$.