METRONMETHOD. REASON. LEARN.
Graphs and coordinates · 12 min

Coordinates and gradient

Graphs show relationships visually. Gradient makes the steepness meaningful as a rate, such as dollars per hour or metres per second.

01Why this matters

Graphs show relationships visually. Gradient makes the steepness meaningful as a rate, such as dollars per hour or metres per second.

02See it

(1, 2)(4, 8)run = 3rise = 6
The line rises six units over a run of three units. Its gradient is 2.
(1, 2)run +3rise +6gradient 2
Read left to right: (1, 2) → run +3 → rise +6 → gradient 2. Each panel connects one part of the example.

03Understand it

Coordinates locate a point using an ordered pair (x, y). Gradient measures how much y changes per unit change in x.

Read the horizontal coordinate x first and the vertical coordinate y second. For two points on a straight line, gradient is change in y divided by change in x. A line rising to the right has positive gradient; a falling line has negative gradient. A vertical line has no defined finite gradient.

04Guided example

Find the gradient between (1, 2) and (4, 8).

  1. The horizontal change is 4 − 1 = 3.
  2. The vertical change is 8 − 2 = 6.
  3. Divide rise by run: m = 6/3 = 2. Each step right increases y by 2.

05Try it

Find gradient from (0, 0) to (3, 9).
A little hint

Divide the rise by the run.

06Practise it

Find gradient from (1, 6) to (3, 2).
A little hint

The rise is negative.

For y = 2x + 3, find y at x = 4.
A little hint

Substitute x into the equation.

07Use it in the real world

On a distance-time graph, a rise of 120 m over a run of 20 s gives a gradient of 6 m/s. The axis units determine the rate’s units.

Go a little deeper

For y = mx + c, m is the gradient and c is the vertical intercept. A line through the origin has c = 0 and expresses direct proportion.

08Quick check

What is the y-intercept of y = 5x + 7?
A little hint

Set x to zero.

09Summary

Coordinates locate a point using an ordered pair (x, y). Gradient measures how much y changes per unit change in x. Divide rise by run: m = 6/3 = 2. Each step right increases y by 2.

Before moving on, explain the worked example in your own words. Use the hint on any question you want to revisit.

10Key idea

m = (y₂ − y₁)/(x₂ − x₁)

m is gradient; x and y are coordinates. Units are y-units per x-unit.

y = mx + c; c = y − mx

Find this in the formula library →