Cambridge IGCSE Mathematics 0580: Coordinates and drawing linear graphs – Study Guide

This study guide teaches two subtopics of Topic 3, Coordinate geometry, in the Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027: 3.1 Coordinates and 3.2 Drawing linear graphs. Both are in the Core (C3.1, C3.2) and Extended (E3.1, E3.2) content. The one difference is in 3.2: Core equations are given as y = mx + c unless a table of values is given, while Extended equations can also come in forms such as y = 7 – 4x or 3x + 2y = 5. That part is labelled Extended only.

Core candidates sit Paper 1 (non-calculator) and Paper 3 (calculator); Extended candidates sit Paper 2 (non-calculator) and Paper 4 (calculator). Either paper can test this content, so practise tables of values by hand.

For the rest of Topic 3, use the Coordinate geometry study guide, revision notes and practice questions. The course is on the Cambridge IGCSE Mathematics hub, and you can tick off every outcome on the printable checklist.

What this unit covers

Syllabus section	What you must be able to do	Tier
C3.1 / E3.1 Coordinates	Use and interpret Cartesian coordinates in two dimensions	Core and Extended
C3.2 / E3.2 Drawing linear graphs	Draw straight-line graphs for linear equations given as y = mx + c (e.g. y = –2x + 5), or from a given table of values	Core and Extended
E3.2 Drawing linear graphs	Draw straight-line graphs for equations in other forms, e.g. y = 7 – 4x and 3x + 2y = 5	Extended only

3.1 Coordinates

The coordinate grid

A Cartesian grid has two number lines that cross at right angles.

• The x-axis is horizontal. The y-axis is vertical.
• They cross at the origin, O, which has coordinates (0, 0).
• A point is written as an ordered pair (x, y). The first number tells you how far to move across (right is positive, left is negative). The second tells you how far to move up or down (up is positive, down is negative).

The order matters. (2, 5) and (5, 2) are different points. A useful memory aid is “along the corridor, then up the stairs”: x first, then y.

The axes split the grid into four regions, called quadrants. Their signs are worth knowing because they let you check a plotted point at a glance:

Region	Sign of x	Sign of y	Example
Top right	+	+	(3, 4)
Top left	–	+	(–3, 4)
Bottom left	–	–	(–3, –4)
Bottom right	+	–	(3, –4)

Points on the axes have a zero coordinate. Every point on the x-axis has y = 0, for example (6, 0). Every point on the y-axis has x = 0, for example (0, –2).

Interpreting coordinates

“Use and interpret” means more than plotting: you read off coordinates, reason about positions, and decide whether a point fits an equation.

• Points with the same x-coordinate lie on a vertical line; points with the same y-coordinate lie on a horizontal line.
• The distance between two such points is the difference of the coordinates that change.
• A point lies on a line if its coordinates make the line’s equation true.

Worked example 1: completing a shape

P(–3, 2), Q(1, 2) and R(1, –4) are three corners of a rectangle PQRS. Find the coordinates of S.

P(-3, 2) and Q(1, 2) share y = 2, so PQ is horizontal.
Q(1, 2) and R(1, -4) share x = 1, so QR is vertical.

In a rectangle, S is opposite Q.
S must be directly below P (same x as P: x = -3)
and level with R (same y as R: y = -4).

S = (-3, -4)

Check: PQ = SR = 1 – (–3) = 4 units and QR = PS = 2 – (–4) = 6 units, so opposite sides match.

Worked example 2: reading coordinates on a scaled grid

Exam grids rarely use one square per unit. Before you read or plot anything, work out what one small square stands for on each axis. The two axes can have different scales.

On a grid, 10 small squares represent 2 units on the x-axis and 5 units on the y-axis. A point A is 7 small squares to the right of the grid line x = 3, and 3 small squares above the grid line y = 10. Find the coordinates of A.

x-axis: 10 small squares = 2 units, so 1 small square = 2 / 10 = 0.2
y-axis: 10 small squares = 5 units, so 1 small square = 5 / 10 = 0.5

x = 3 + 7 x 0.2 = 3 + 1.4 = 4.4
y = 10 + 3 x 0.5 = 10 + 1.5 = 11.5

A = (4.4, 11.5)

Count from a labelled grid line close to the point, not from the origin, to cut counting slips.

Worked example 3: does a point lie on a line?

Does the point (4, –7) lie on the line y = –3x + 4?

Substitute x = 4:  y = -3(4) + 4 = -12 + 4 = -8
The line passes through (4, -8), not (4, -7).
So (4, -7) does not lie on the line.

The point (4, –7) is one unit above the line. The same substitution is how you check each point in a table of values before you plot it.

3.2 Drawing linear graphs

A linear equation is one where x and y appear only to the power 1, with no xy term. Its graph is always a straight line. So you only need two correct points to fix the line, but you should always work out three. The third point is a check: if the three plotted points do not line up, one of them is wrong.

Method: drawing a straight line from its equation

1. Read the range of x-values you are given, for example –2 ≤ x ≤ 3.
2. Choose at least three x-values, including the two end values of the range. Include x = 0 if it lies in the range, because it is quick to work out.
3. Substitute each x-value into the equation to find y. Write the results in a table.
4. Plot each point with a small, neat cross on the grid.
5. Join the points with a single straight line using a ruler. Draw it across the whole of the given range, not just between your plotted points.
6. Label the line with its equation if more than one line is drawn on the same grid.

Worked example 4: y = mx + c with a negative gradient (Core and Extended)

Draw the graph of y = –3x + 4 for –2 ≤ x ≤ 3.

Work out the table, taking care with the negative signs. Brackets around negative x-values stop most sign slips.

x = -2:  y = -3(-2) + 4 =  6 + 4 = 10
x = -1:  y = -3(-1) + 4 =  3 + 4 =  7
x =  0:  y = -3(0)  + 4 =  0 + 4 =  4
x =  1:  y = -3(1)  + 4 = -3 + 4 =  1
x =  2:  y = -3(2)  + 4 = -6 + 4 = -2
x =  3:  y = -3(3)  + 4 = -9 + 4 = -5

x	–2	–1	0	1	2	3
y	10	7	4	1	–2	–5

Plot (–2, 10), (–1, 7), (0, 4), (1, 1), (2, –2) and (3, –5) and rule a line from x = –2 to x = 3. The y-axis must reach from at least –5 up to 10, so check the grid has room before you start.

Check the table: each time x goes up by 1, y goes down by 3. A value that breaks this steady pattern is the one to recalculate. At x = 0, y equals c (here 4), where the line crosses the y-axis.

Worked example 5: completing a given table (Core and Extended)

When a table of values is printed for you, complete it, then plot every point in it. You may be asked to use a table even when the gradient is a fraction.

Complete the table for y = ½x – 1 and draw the graph for –4 ≤ x ≤ 4.

x	–4	–2	0	2	4
y		–2		0

x = -4:  y = 1/2 x (-4) - 1 = -2 - 1 = -3
x =  0:  y = 1/2 x 0    - 1 =  0 - 1 = -1
x =  4:  y = 1/2 x 4    - 1 =  2 - 1 =  1

x	–4	–2	0	2	4
y	–3	–2	–1	0	1

The y-values go up by 1 each time x goes up by 2, so the pattern holds. Plot all five points and join them with one ruled line from x = –4 to x = 4. The line crosses the x-axis at (2, 0).

The table uses even x-values so that halving gives whole numbers. Make the same choice when you pick your own x-values for a fractional gradient.

Extended only: equations in other forms

Extended papers can give the equation with the x-term second, such as y = 7 – 4x, or with x and y on the same side, such as 3x + 2y = 5. The graph is still a straight line, so the method is the same: find three points, plot, rule.

Worked example 6 (Extended only): y = 9 – 2x for 0 ≤ x ≤ 5

Substitute straight in. The equation is the same as y = –2x + 9; only the order of the terms differs.

x = 0:  y = 9 - 2(0) = 9
x = 2:  y = 9 - 2(2) = 5
x = 5:  y = 9 - 2(5) = -1

Plot (0, 9), (2, 5) and (5, –1), and rule the line. A common slip is to read “9 – 2x” as “(9 – 2) x”, giving 7x. Multiplication is done before subtraction.

Worked example 7 (Extended only): 4x + 3y = 12

When x and y are on the same side, the fastest method is to substitute x = 0 and y = 0 in turn. This gives the two points where the line crosses the axes.

x = 0:  3y = 12        so y = 4    point (0, 4)
y = 0:  4x = 12        so x = 3    point (3, 0)
check, x = 6:  24 + 3y = 12
               3y = -12  so y = -4   point (6, -4)

Plot (0, 4), (3, 0) and (6, –4). They lie on one straight line, so the line can be drawn.

Worked example 8 (Extended only): 5x – 2y = 10 for –1 ≤ x ≤ 4

Here the range is given, so the end values x = –1 and x = 4 must be in the table. It is easier to rearrange first.

5x - 2y = 10
     -2y = 10 - 5x
       y = (5x - 10) / 2

x = -1:  y = (-5 - 10) / 2 = -15 / 2 = -7.5
x =  0:  y = (0 - 10) / 2  = -5
x =  2:  y = (10 - 10) / 2 = 0
x =  4:  y = (20 - 10) / 2 = 5

Plot (–1, –7.5), (0, –5), (2, 0) and (4, 5), and rule the line across –1 ≤ x ≤ 4. Dividing by –2 flips the signs of both terms, which is where most errors happen.

Common errors

• Swapping the coordinates. Plotting (y, x) instead of (x, y), for example putting (3, –5) at 5 left and 3 up, which is (–5, 3).
• Sign slips with negative x. Working out –3 × –2 as –6 instead of 6. Put negative numbers in brackets when you substitute.
• Misreading the scale. Assuming each small square is 1 unit when it is 0.2 or 0.5. Work out the scale on both axes before plotting.
• Stopping the line short. Drawing only between the plotted points instead of across the full range given in the question.
• Only two points. With no third point, one wrong value gives a wrong line with nothing to warn you. If your points are not in a line, fix the table rather than bending the line.
• (Extended only) Rearranging wrongly. Turning 4x + 3y = 12 into y = 12 – 4x by forgetting to divide by 3, or losing a minus sign when dividing by a negative.

Where this leads

Reading and drawing lines are the starting skills for gradient (3.3), equations of lines (3.5) and parallel lines (3.6), and for solving equations graphically in Topic 2. Next, work through the Coordinate geometry practice questions. To find which parts of the course need work, try the Core diagnostic or the Extended diagnostic. Other resources for syllabus 0580 are on the hub.

Official syllabus

Cambridge International, Cambridge IGCSE Mathematics 0580 syllabus for 2025, 2026 and 2027, Subject content, Topic 3 Coordinate geometry: C3.1 and E3.1 Coordinates; C3.2 and E3.2 Drawing linear graphs.