Sheet № 25 · Foundation + Higher · AQA · Edexcel · OCR
Linear Graphs & Equation of a Line –
Linear graphs are one of the most frequently tested topics across AQA, Edexcel and OCR GCSE Maths papers. Whether you are on Foundation or Higher tier, you need to plot straight lines, read off gradients, identify intercepts and write the equation of a line. This guide walks you through everything from the basics of y = mx + c to finding
§Key definitions
Parallel lines
have the same gradient: m₁ = m₂
Perpendicular lines
have gradients that multiply to −1:
Question:
A line passes through (0, 4) and (3, 10). Find its equation.
Step 1:
The line passes through (0, 4), so the y-intercept c = 4.
Step 2:
Calculate the gradient.
§Formulas to memorise
y = mx + c
m = (y₂ − y₁) / (x₂ − x₁)
m₁ × m₂ = −1
Parallel lines: have the same gradient: m₁ = m₂
Perpendicular lines: have gradients that multiply to −1:
Identify m and c — from the equation. For example, y = 2x + 3 has m = 2 and c = 3.
Plot the y-intercept — (0, c) on the y-axis.
Use the gradient — to find a second point. Gradient 2 means go 1 right and 2 up.
Plot a third point — the same way to confirm the line is straight.
Draw a straight line — through the points using a ruler, extending it across the grid.
Worked example
A line passes through (0, 4) and (3, 10). Find its equation.
Step 1: The line passes through (0, 4), so the y-intercept c = 4.
⚠ Common mistakes
- ✗Mixing up rise and run. The gradient is change in y divided by change in x, not the other way round. Always put the vertical change on top.
- ✗Forgetting the sign of the gradient. If the line slopes downward from left to right, the gradient is negative. Students often write it as positive.
- ✗Using the wrong reciprocal for perpendicular lines. The perpendicular gradient is the negative reciprocal, not just the reciprocal. If m = 2, the perpendicular gradient is −½, not ½.
- ✗Not reading the scale carefully. On exam graphs, the axes may not go up in ones. Always check what each square represents.
- ✗Writing x = instead of y =. The equation of a straight line is y = mx + c. Lines of the form x = k are vertical lines, which have undefined gradient.
✦ Exam tips
- →Show your gradient calculation. Write out (y₂ − y₁) / (x₂ − x₁) in full — examiners award method marks even if you make an arithmetic slip.
- →Label graphs clearly. When plotting, always label the line with its equation. On AQA papers in particular, the mark scheme often requires this.
- →Use the cover-up method for lines in the form ax + by = c. Set x = 0 to find where it crosses the y-axis, then set y = 0 for the x-axis crossing.
- →For perpendicular questions, write the word "negative reciprocal" in your working — it shows the examiner you understand the concept and can earn you an extra communication mark on Edexcel papers.