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MathematicsGCSE, IGCSE & A Level

What is gradient?

The gradient of a line measures its steepness — the change in y divided by the change in x between two points.

Gradient explained

A positive gradient rises left to right, a negative gradient falls, and a horizontal line has gradient zero. Parallel lines share a gradient; perpendicular gradients multiply to −1-1.
For a curve, the gradient changes from point to point and is found from the tangent, or exactly by differentiation at A Level.

Key formula

m=y2−y1x2−x1m = \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}

Worked example

Through (1,2)(1, 2) and (4,11)(4, 11): m=11−24−1=3m = \frac{11 - 2}{4 - 1} = 3, so the line is y=3x−1y = 3x - 1.

Examiner tip

In context questions, always give the gradient units — pounds per hour, metres per second — because the interpretation mark depends on them.

Gradient: common questions

What is the gradient of a vertical line?
It is undefined, because the change in x is zero and division by zero has no value.
How do I find the gradient of a curve at a point?
At GCSE, draw a tangent and use rise over run. At A Level, differentiate and substitute the x-coordinate.

Related terms

Differentiation

Differentiation is the process of finding the rate at which one quantity changes with respect to another — the gradient of a curve at a point.

Stationary point

A stationary point is a point on a curve where the gradient is zero — a maximum, a minimum or a point of inflection.

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