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Gradient and straight-line graphs

Years 8 to 11

The gradient of a line tells you how steep it is, and y = mx + c describes it.

The line y = 2x + 1. For every 1 step along, it goes up 2. It crosses the y-axis at 1.

For learners

The gradient tells you how steep a line is. Go along 1 step, then see how far the line goes up. That is the gradient.

A line with the rule $y = 2 x + 1$ goes up 2 each time you move 1 along. It crosses the up-and-down axis at 1.

In $y = m x + c$ , the number $m$ is the gradient and $c$ is where the line crosses the y-axis.

A worked example

Find the gradient of the line through (1, 3) and (3, 7).

Change in $y$ : 7 − 3 = 4.
Change in $x$ : 3 − 1 = 2.
Gradient = 4 ÷ 2 = 2.

For grown-ups

Gradient = change in $y$ ÷ change in $x$ (rise over run). Parallel lines have the same gradient. A line that goes down as you move right has a negative gradient.

Common mistakes

Dividing the change in $x$ by the change in $y$ (run over rise).
Losing the sign when the line slopes downwards.
Reading the intercept from the x-axis instead of the y-axis.

Why it matters

Gradients describe speed on a distance-time graph, the rate of change in science, and lead to calculus.

Topics in quarter that use this

Gradient of a straight line
Equation of a straight line
Quadratic graphs