Free GCSE Maths lesson: Algebra
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Lesson 34 · GCSE / Key Stage 4 · Maths · Algebra

Coordinates and Straight-Line Graphs

Learn how to plot points, draw straight lines accurately, and connect a line's equation to its gradient and intercept.

Qualification: GCSE Key Stage 4 Subject: Maths Strand: Algebra

Lesson overview

A core algebra and graph topic for every GCSE Maths board.

Straight-line graphs turn algebra into a picture. You need to read coordinates, plot points accurately, draw lines from equations, and interpret gradient as a rate of change.

QualificationGCSE Mathematics
Key stageKey Stage 4
StrandAlgebra
Tier Both
Calculator status Calculator and non-calculator
Exam-board status Shared skill

Visual summary

Before viewing

Before viewing, point to the gradient, y-intercept and one plotted coordinate.

Infographic explaining GCSE Maths coordinates and straight-line graphs, including plotting points, tables of values, gradient, y-intercept, parallel lines and axis checks.
Use this visual to connect coordinates, tables of values, gradient, y = mx + c, parallel lines and common graph accuracy checks.Download visual

After viewing

After viewing, identify m and c in one line equation and sketch its starting point.

What you will learn

  • How to read and plot coordinates in all four quadrants.
  • How to recognise horizontal and vertical lines.
  • How to build a table of values for a linear equation.
  • How to draw a straight-line graph accurately.
  • How gradient describes steepness, direction and rate of change.
  • How y = mx + c connects an equation to a graph, intercept and parallel lines.
  • How to use a shared gradient and one point to find the equation of a parallel line.

Key facts before you start

Coordinate order Coordinates are written as (x, y): across first, then up or down.
Gradient Gradient measures steepness: change in y divided by change in x.
Intercept The y-intercept is where the graph crosses the y-axis.
Line equation y = mx + c shows gradient m and intercept c.

Why this matters

Graphs appear across algebra, ratio, compound measures, science and real-life data. If you can move between an equation, a table and a graph, many GCSE questions become more manageable.

The key habit is accuracy: read the x-value first, choose a sensible scale, plot points carefully and check that the line matches the equation.

Prior knowledge

You should already be comfortable with:

  • using negative numbers on a number line,
  • substituting values into simple formulae,
  • solving one-step equations,
  • reading scales on axes,
  • spotting linear patterns.

Clear explanation

Coordinates are written as (x, y)

The x-coordinate tells you how far to move across. The y-coordinate tells you how far to move up or down. For (3, −2), move 3 right and 2 down.

Coordinate grid showing y = 2x + 1 A labelled coordinate grid with the straight line y equals two x plus one passing through minus one minus one, zero one, one three and two five. x y 0 (0, 1) (1, 3) (2, 5) Line: y = 2x + 1
Checked diagram: the line y = 2x + 1 passes through (0, 1), (1, 3) and (2, 5).

Use a table of values to draw a line

For y = 2x + 1, choose some x-values, substitute each one, then plot the coordinate pairs.

x: −1, 0, 1, 2 y = 2x + 1 gives −1, 1, 3, 5 Coordinates: (−1, −1), (0, 1), (1, 3), (2, 5)

Gradient means rise divided by run

The gradient tells you how steep the line is. A positive gradient slopes upwards from left to right. A negative gradient slopes downwards.

Gradient = change in y ÷ change in x From (0, 1) to (2, 5): change in y = 4 and change in x = 2 Gradient = 4 ÷ 2 = 2

In y = mx + c, m and c have jobs

In y = mx + c, m is the gradient and c is the y-intercept. The y-intercept is where the line crosses the y-axis, so it is the y-value when x = 0.

Worked examples

Example 1: Plot a coordinate

Describe how to plot (−2, 4).

Move 2 left from the origin, then 4 up.
Reveal answer

Answer: The point is in the top-left quadrant.

Example 2: Complete a table of values

Complete y = 3x − 2 for x = 0, 1, 2.

x = 0: y = −2 x = 1: y = 1 x = 2: y = 4
Reveal answer

Answer: (0, −2), (1, 1), (2, 4).

Example 3: Find a gradient

Find the gradient of the line through (1, 2) and (4, 8).

Change in y = 8 − 2 = 6 Change in x = 4 − 1 = 3
Reveal answer

Answer: Gradient = 6 ÷ 3 = 2.

Example 4: Read y = mx + c

For y = −4x + 7, state the gradient and y-intercept.

Reveal answer

Answer: Gradient = −4 and y-intercept = 7.

Quick checks

Choose an answer, then check your thinking.

1. In the coordinate (5, −3), what is the x-coordinate?

2. What is the y-intercept of y = 4x − 6?

3. A line goes through (0, 2) and (3, 8). What is its gradient?

Practice questions

Question 1

Write the x-coordinate and y-coordinate of (−4, 7).

Reveal answer and marking guidance

Answer: x = −4 and y = 7.

Marking: Use the order (x, y). The first number is x and the second number is y.

Question 2

For y = 2x − 3, find y when x = 5.

Reveal answer and marking guidance

Answer: y = 7.

Marking: Substitute x = 5: y = 2 × 5 − 3 = 7.

Question 3

Complete the coordinates for y = x + 4 when x = −2, 0 and 3.

Reveal answer and marking guidance

Answer: (−2, 2), (0, 4), (3, 7).

Marking: Substitute each x-value into y = x + 4 and write each result as a coordinate pair.

Question 4

Find the gradient of the line through (2, 3) and (6, 11).

Reveal answer and marking guidance

Answer: 2.

Marking: Change in y = 11 − 3 = 8 and change in x = 6 − 2 = 4, so gradient = 8 ÷ 4 = 2.

Question 5

State the gradient and y-intercept of y = −3x + 5.

Reveal answer and marking guidance

Answer: Gradient = −3 and y-intercept = 5.

Marking: Compare the equation with y = mx + c. The coefficient of x is the gradient and the constant is the y-intercept.

Question 6

A straight line has gradient 4 and crosses the y-axis at −1. Write its equation.

Reveal answer and marking guidance

Answer: y = 4x − 1.

Marking: Put m = 4 and c = −1 into y = mx + c.

Question 7

Find the equation of the line with gradient 12 that passes through (0, −3).

Reveal answer and marking guidance

Answer: y = 12x − 3.

Marking: Use y = mx + c with m = 12 and c = −3 because the point (0, −3) is the y-intercept.

Question 8

A line is parallel to y = −2x + 4 and passes through (0, 7). Write its equation.

Reveal answer and marking guidance

Answer: y = −2x + 7.

Marking: Parallel lines have the same gradient, so m = −2. The y-intercept is 7 because the line passes through (0, 7).

Question 9

Find the equation of the straight line through (−2, 5) and (4, −7).

Reveal answer and marking guidance

Answer: y = −2x + 1.

Marking: Change in y = −7 − 5 = −12 and change in x = 4 − (−2) = 6, so gradient = −2. Substitute (−2, 5): 5 = −2(−2) + c, so c = 1.

Question 10

A line is parallel to y = 3x − 4 and passes through (2, 5). Find its equation.

Reveal answer and marking guidance

Answer: y = 3x − 1.

Marking: Parallel lines have the same gradient, so m = 3. Substitute (2, 5) into y = 3x + c: 5 = 3 × 2 + c, so c = −1.

Answers and marking guidance

The exact practice answers are hidden under each question so you can try first. For coordinates and straight-line graphs, marks usually come from using the correct coordinate order, substituting x-values accurately, plotting points with a sensible scale, showing a table of values, calculating gradient as change in y divided by change in x, and using a known point to find the y-intercept when it is not given directly.

Common mistakes

  • Swapping coordinates: (−4, 7) is not the same point as (7, −4).
  • Using uneven axis scales: each square must represent the same step on one axis unless the scale is clearly labelled.
  • Forgetting negative signs: y = −3x + 5 has a negative gradient, so it slopes down from left to right.
  • Reading c as the x-intercept: in y = mx + c, c is the y-intercept.
  • Drawing through one point only: use at least two correct points, and preferably three as a check.

Extension challenge

A line is parallel to y = 3x − 2 and passes through (0, 6). Write its equation.

Reveal answer

Answer: y = 3x + 6.

Parallel lines have the same gradient. The y-intercept is 6 because the line passes through (0, 6).

Exam-board guidance

Coordinates and straight-line graphs are assessed by every GCSE Maths board. Questions may ask for plotted points, tables of values, graph drawing, gradients, intercepts, equations of lines or practical graph interpretation.

AQA GCSE Maths

Focus on the shared method, notation and checking habit; wording and context can vary by route.

OCR GCSE Maths

Show clear method steps and use precise notation; question wording may vary by route.

Pearson Edexcel GCSE Maths

Keep working visible and check units or notation; contexts may vary by route.

Eduqas GCSE Maths

Use the same core skill in practical contexts, with clear reasoning and final checks.

WJEC Wales

Connect the method to the context and state what the result means.

CCEA GCSE Maths

Make each method step visible and keep notation, units and final checks clear.

Next lesson

Next, move into Simultaneous Equations.

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