Gradient of a Line

The gradient (slope) of a line measures how steep it is — how much the line rises or falls for every unit it moves to the right. Gradient = change in y ÷ change in x (rise ÷ run).

What is Gradient?

The gradient (or slope) of a line tells you how steep it is — how much it rises or falls for each unit it moves to the right.

$\text{Gradient} = \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Positive, Negative, Zero, and Undefined Gradients

Gradient	What it means	Example $m$
Positive	Line goes uphill (left to right)	$m = 2$
Negative	Line goes downhill (left to right)	$m = -3$
Zero	Horizontal line	$m = 0$
Undefined	Vertical line	division by zero

Calculate the gradient between (2, 1) and (6, 9)

$m = \frac{9 - 1}{6 - 2} = \frac{8}{4} = 2$

For every 1 step right, the line goes up 2.

Gradient of a downhill line — (1, 8) and (5, 2)

$m = \frac{2 - 8}{5 - 1} = \frac{-6}{4} = -1.5$

Negative gradient means the line falls from left to right.

Gradient from a Graph

Pick two clear grid points on the line. Draw a right-angled triangle underneath:

Horizontal leg = run ( $\Delta x$ )
Vertical leg = rise ( $\Delta y$ )

Then $m = \Delta y \div \Delta x$ .

Warning

Always go from left to right when reading the rise and run to avoid sign errors.

Connecting to $y = mx + c$

The gradient $m$ in the equation $y = mx + c$ is exactly the gradient calculated from any two points on the line.

Real world: road gradient signs

A road sign showing 10 % gradient means the road rises 10 m for every 100 m of horizontal distance — a gradient of $m = 0.1$ . Lorry drivers shift to a lower gear on slopes steeper than about 7 %.

Real world: wheelchair ramp regulations

Building codes specify a maximum ramp gradient of 1 : 12 (≈ 0.083). A ramp rising 0.5 m must therefore extend at least $0.5 \times 12 = 6$ m horizontally to be safe and accessible.

Real world: ski slope difficulty

A green ski run might have gradient 0.15 (gentle), while a black run can exceed 0.6 (very steep). Gradient is the key number that classifies difficulty for skiers.

Practice Problems

Find the gradient between (0, 3) and (4, 11).

Show answer

m = (11−3)/(4−0) = 8/4 = 2

A line passes through (−2, 5) and (3, −5). Find its gradient.

Show answer

m = (−5−5)/(3−(−2)) = −10/5 = −2

What is the gradient of a horizontal line that passes through (0, 4) and (7, 4)?

Show answer

m = (4−4)/(7−0) = 0. Horizontal lines always have gradient 0.

Gradient of a Line

What is Gradient?

Positive, Negative, Zero, and Undefined Gradients

Gradient from a Graph

Connecting to y=mx+cy = mx + cy=mx+c

Practice Problems

Connecting to $y = mx + c$