Gradient of a Line

What is a gradient?

The gradient (m) — also called the slope in American usage — measures how much a line "rises" or "falls" relative to the x-axis. Mathematically: m = change in y / change in x = Δy / Δx. If you move 1 unit to the right along the line, the gradient tells you how far up (or down) you move on the vertical axis.

To find the gradient between two points (x₁, y₁) and (x₂, y₂), use the formula m = (y₂ − y₁) / (x₂ − x₁). The order matters: the numerator and denominator must use the same point first — if you take y₂ in the numerator, x₂ must appear in the denominator.

The four types of gradient

• Positive gradient (m > 0): the line "rises" — as you move right, y increases.
• Negative gradient (m < 0): the line "falls" — as you move right, y decreases.
• Zero gradient (m = 0): the line is horizontal — y stays constant. For example (1,3)→(5,3).
• Undefined gradient: the line is vertical, x₁ = x₂. You cannot divide by 0, so the gradient is undefined.

Three worked examples

1. (1, 2) and (4, 5): m = (5 − 2) / (4 − 1) = 3 / 3 = 1. Positive gradient, the line rises by 1 for every unit to the right.
2. (0, 10) and (5, 0): m = (0 − 10) / (5 − 0) = −10 / 5 = −2. Negative gradient, the line falls by 2 for every unit to the right.
3. (0, 0) and (2, 1): m = (1 − 0) / (2 − 0) = 1/2 = 0.5. The gradient is a half — for every 2 units to the right, it rises by 1 unit.

Common mistakes

• Swapping the order! Calculating (y₁ − y₂) / (x₂ − x₁) instead of (y₂ − y₁) / (x₂ − x₁) gives a gradient with the wrong sign.
• Mixing up numerator and denominator. Remember: y is the "rise", x is the "run". Gradient = rise over run.
• Not spotting an undefined gradient. If x₁ = x₂, don't write "m = 0" — that is a vertical line and the gradient is undefined!
• Getting negative signs wrong. When subtracting a negative number, the sign flips: 3 − (−2) = 5.