Mathématiques

What Is Gradient in Maths?

Gradient is the change in y divided by the change in x. Worked from two coordinates, with the sign rule and why a vertical line's gradient is undefined.

Gradient

A measure of how steep a line is, found by dividing the change in y by the change in x between any two points on it.

Aussi appelé: Slope
Où les élèves le rencontrent: Grade 7 or 8 mathematics, with straight-line graphs, and again on the Digital SAT where the same quantity is always called the slope.

La réponse en bref
Gradient measures how steep a line is: the change in y divided by the change in x between any two points on it, usually said as rise over run. A gradient of 3 means the line rises 3 units for every 1 across. A negative gradient falls as you move right.

Un exemple

(1, 4) and (4, 13): (13 − 4) / (4 − 1) = 9 / 3 = 3

Going from the first point to the second, y climbs by 9 while x moves across by 3, so the line rises 3 units for every 1 unit of horizontal travel. Taking the points the other way round gives (4 − 13) / (1 − 4) = −9 / −3, which is still 3 — the two minus signs cancel. That is worth testing once, because it is the reason the order of the points does not matter, provided you keep the same order on the top and the bottom.

From two coordinates to a number

The formula is (y₂ − y₁) / (x₂ − x₁), and every part of it is the phrase rise over run written in symbols. Subtract the y values to get the rise, subtract the x values in the same order to get the run, divide.

The one rule that matters is consistency. Reverse both subtractions and the two sign changes cancel out, leaving the answer untouched. Reverse only one of them and you get the right number with the wrong sign, which turns an uphill line into a downhill one and is very hard to spot afterwards.

What the number is telling you

Two separate pieces of information sit in one value. The sign says which way the line leans: positive rises as you read left to right, negative falls. The size says how sharply — a gradient of 3 is three times as steep as a gradient of 1, and a gradient of 1/3 is a gentle climb.

In the form y = mx + c the gradient is m, sitting in front of the x, but only once the equation is in that form. y = 5 − 2x has a gradient of −2, not 5 and not 2, and reading the first number off the page is the most common way to lose the mark. Rearrange first, then read.

Zero and undefined are not the same answer

A horizontal line such as y = 2 has a rise of 0 over any run you choose, so its gradient is 0 ÷ 5 = 0. Flat means zero, and zero means flat.

A vertical line such as x = 2 is the opposite case and it is not the mirror image people expect. The run is 0, so the calculation asks you to divide by zero, and the gradient is undefined. Not zero, not infinite — undefined, which is the answer an examiner is looking for and the reason vertical lines cannot be written as y = mx + c at all.

One extension is worth knowing before you meet it. Curves do not have a single gradient, because their steepness changes from point to point. When a question asks for the gradient of a curve it means the gradient at one named point, found from the straight line that just grazes the curve there.

Questions fréquentes

Is gradient the same as slope?

Yes — the same quantity with two names. British specifications say gradient, American ones say slope, and the Digital SAT uses slope throughout. A student sitting IGCSE Mathematics and the SAT in the same year will see both words for the same calculation and should treat them as interchangeable.

Does it matter which point you subtract first?

Not as long as you keep the same order top and bottom. Both subtractions reversed gives two sign changes that cancel, so the answer is identical. Only one reversed leaves the size right and the sign wrong. Writing the two coordinates down in a fixed order before starting removes the risk.

What is the gradient of a vertical line?

Undefined. The horizontal change between any two points on it is zero, and dividing by zero is not something you are allowed to do. This is different from a horizontal line, whose gradient is genuinely zero, and the two cases are worth keeping apart because papers ask about them in the same breath.

How do you find the gradient from an equation like 2x + 4y = 12?

Rearrange into y = mx + c first. Subtract 2x to get 4y = 12 − 2x, then divide by 4 to get y = 3 − 0.5x, so the gradient is −0.5. Trying to read a gradient off the original form is guesswork, and the minus sign is the part most often lost.

Dernière mise à jour 21 août 2026