CAMBRIDGE O LEVEL · 2025–2027

Gradient of linear graphs.

Measure the vertical change for each horizontal unit, including its sign.

Syllabus 3.3Visual notesWorked examples

BY THE END

Read and calculate gradients · use two coordinates consistently · recognise special cases

Gradient measures signed steepness

Gradient measures how much a straight line rises or falls for a given horizontal move. A line that rises 2 units for every 1 unit right has gradient 2.

Build a right-angled step between two points on the line. Compare the vertical change with the horizontal change by dividing rise by run. Keep the direction consistent: travelling right along a falling line gives a negative vertical change and therefore a negative gradient.

Gradient is the vertical change divided by the horizontal change. Read a line from left to right: rising means positive; falling means negative.

m = change in ychange in x = y₂ − y₁x₂ − x₁

GRADIENT LAB · KEEP THE RUN AT 4

Move 4 units right from P; the vertical change is 2. Gradient is 2 divided by 4.

Line through P and Q

m = 24 = 0.5

Rises from left to right: positive gradient.

Pause and explainWhat does a gradient of −2 mean?

For each 1-unit move to the right, the line goes down 2 units.

Use two points in the same order

Coordinates let you calculate the same rise and run without measuring a drawing. Subtract the starting y-value from the ending y-value for the vertical change, and do the matching subtraction for x.

Use the same point order in numerator and denominator. Reversing both differences is harmless because both signs change, but reversing just one gives the wrong sign. A vertical line has zero horizontal change, so its gradient is undefined rather than zero.

WORKED EXAMPLE

Find the gradient through A (−2, 5) and B (4, −1).

m = −1 − 54 − (−2) = −66 = −1

Reversing both differences also works: (5 − (−1)) ÷ (−2 − 4) = 6 ÷ (−6) = −1.

Horizontal line

y does not change, so the gradient is 0.

Vertical line

x does not change. Division by zero is undefined, so its gradient is undefined, not zero.

Pause and explainWhich kind of line has gradient zero?

A horizontal line. Its vertical change is zero while its horizontal change can be non-zero.

Your turn. Rise over run.

Sketch the direction of the line before calculating so you can predict whether its gradient is positive, negative, zero or undefined.

Label your two points and use the same subtraction order for both coordinate differences. Simplify the resulting fraction and compare the sign with your sketch.

Find the gradient through (1, −2) and (5, 6).

Reveal worked answer

(6 − (−2))/(5 − 1) = 84 = 2.

Find the gradient through (−4, 3) and (2, 1).

Reveal worked answer

(1 − 3)/(2 − (−4)) = −26 = −13.

State the gradients of y = 5 and x = −3.

Reveal worked answer

y = 5: gradient 0. x = −3: gradient undefined.