Coordinates and linear graphs

Revise Coordinates and linear graphs for Mathematics D 4024 (O Level) — revision notes, 12 practice questions and instant AI marking.

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Cambridge O Level Mathematics (Syllabus D) 4024

Coordinates and linear graphs

The big idea: Coordinates fix a point by how far across and how far up it is. A linear equation is a rule linking x and y, and every point that obeys the rule lies on one straight line.

Summary — What This Chapter Covers

Use and interpret Cartesian coordinates (x, y) in all four quadrants.
Draw straight-line graphs from equations such as y = −2x + 5, y = 7 − 4x and 3x + 2y = 5.
Recognise horizontal and vertical lines, and test whether a point lies on a line.

1. Cartesian Coordinates

Syllabus 3.1 · Paper 1 & 2

Across first, then up or down

A point (x, y) is x units across from the origin (right is positive, left is negative) and y units up (up is positive, down is negative). The axes divide the plane into four quadrants. Every point on the x-axis has y = 0; every point on the y-axis has x = 0.

Practice Question 1

A(1, 2), B(5, 2) and C(5, −1) are three corners of a rectangle ABCD. Find the coordinates of D.

Practice Question 2

P(−2, 1), Q(2, 3) and R(4, −1) are three vertices of the parallelogram PQRS. Find S.

2. Drawing a Straight Line from Its Equation

Syllabus 3.2 · Paper 1 & 2

A table of three points

Choose three x-values, work out y for each, plot the points and join them with a ruler, extending across the whole grid given. Three points (not two) let you spot an arithmetic slip: they must lie in a straight line. For ax + by = c, the easiest points are the intercepts (put x = 0, then y = 0), or rearrange to make y the subject.

Worked Example

Complete a table of values for y = −2x + 5 for x = −1, 0, 1, 2, 3.

Practice Question 3

(a) Find three points on 3x + 2y = 5 with whole-number coordinates, using x = −1, 1 and 3. (b) Does the point (4, −3) lie on y = 7 − 4x?

3. Horizontal and Vertical Lines

Syllabus 3.2 · Paper 1 & 2

One coordinate stays fixed

x = k y = c

Practice Question 4

Write down the equation of (a) the vertical line through (3, −2) (b) the horizontal line through (3, −2) (c) the point where these two lines meet.

Memorise

Order

Axes

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Coordinates and linear graphs