Gradient of a Line

Grade 4

What gradient means

The gradient of a line tells you how steep it is — more precisely, how much the y-value changes for every 1 unit you move to the right along x.

A gradient of 3 means: go 1 across, go 3 up. A gradient of −2 means: go 1 across, go 2 down. A gradient of 0 is a flat horizontal line — no change at all.

How to calculate it

Pick any two points on the line. The gradient is the rise (change in y) divided by the run (change in x):

Gradient = (y₂ − y₁) ÷ (x₂ − x₁)
It doesn't matter which point you call (x₁, y₁) and which you call (x₂, y₂) — as long as you're consistent. Flip both and you get the same answer.

Worked example — from a graph

EXAMPLE
A straight line passes through the points (1, 3) and (5, 11). Find the gradient.
Rise = 11 − 3 = 8. Run = 5 − 1 = 4. Gradient = 8 ÷ 4 =2.

Positive, negative, zero

A positive gradient slopes upward left to right. A negative gradient slopes downward. A zero gradient is horizontal. A vertical line has undefined gradient — the run is zero and you can't divide by zero.

When reading a gradient from a graph, use the scale on both axes to measure rise and run — don't just count squares unless the grid is 1:1. A stretched axis will give you the wrong answer if you count squares without checking.
Nova
Try this one

The diagram shows a distance-time graph for two runners, A and B, in a race.

t (s)d (m)0102030404080120160ABNot drawn accurately

(a) Find the speed of runner A. [2]

(b) Find the speed of runner B. [1]

(c) How much further does runner A travel than runner B in 40 seconds? [1]

Nova's hint:

This question uses the gradient of a distance-time graph to find speed — speed = distance ÷ time.

Ask yourself: 'What are the start and end coordinates for each line, and how do I use these to find gradient?'

Then subtract the two distances at t = 40 to find the difference.

Try to Solve it with Nova?