Straight-Line Graphs (y = mx + c)

GCSE Maths · Algebra

Straight-line graphs

A straight line has the equation y = mx + c, where:

The gradient tells you how steep the line is and in which direction it slopes.

gradient = change in y ÷ change in x  =  rise ÷ run

A positive gradient slopes up (left to right); a negative gradient slopes down.
A larger number = steeper line.

Between two points (x₁, y₁) and (x₂, y₂):

m = (y₂ − y₁) ÷ (x₂ − x₁)

This is the value of c — where the line cuts the y-axis (when x = 0). In y = 3x − 2, the line crosses the y-axis at −2.

For y = 4x + 1: gradient m = 4, y-intercept c = 1.

Rearrange first if needed: 2y = 6x + 8 → divide by 2 → y = 3x + 4 (m = 3, c = 4).

Parallel lines have the same gradient (m).
Perpendicular lines have gradients whose product is −1 (the negative reciprocal). If one gradient is 2, the perpendicular gradient is −½.

1. Mark the y-intercept (c) on the y-axis.

2. Use the gradient (rise/run) to find another point.

3. Join them with a straight line.

Or make a small table of values: pick x-values, work out y, plot the points.

Given a gradient and a point, substitute into y = mx + c to find c. E.g. gradient 2 through (1, 5): 5 = 2(1) + c → c = 3 → y = 2x + 3.

Find the gradient of the line through (2, 1) and (6, 9).

m = (9 − 1) ÷ (6 − 2) = 8 ÷ 4 = 2

✓

Reading m and c from an equation that isn't in y = mx + c form (rearrange first).
Getting the gradient formula upside down — it's change in y over change in x.
Forgetting perpendicular gradients are negative reciprocals (product −1).

Always rearrange to y = mx + c before reading off gradient and intercept.
For "parallel", copy the gradient; for "perpendicular", use the negative reciprocal.
Label at least two clear points when finding a gradient from a graph.

y = mx + c: m = gradient (rise ÷ run), c = y-intercept.
Gradient between points = (y₂ − y₁) ÷ (x₂ − x₁).
Parallel = same gradient; perpendicular = gradients multiply to −1 (negative reciprocal).