What is gradient?
The gradient of a straight line tells us how steep it is and which way it slopes. It compares how far the line rises (up or down) with how far it moves across.
Key idea
Gradient = vertical change ÷ horizontal change. For two points (x₁, y₁) and (x₂, y₂): gradient = (y₂ − y₁) ÷ (x₂ − x₁).
Finding the gradient from two points
Worked example
Find the gradient of the line through (1, 2) and (3, 8).
Gradient = (8 − 2) ÷ (3 − 1) = 6 ÷ 2 = 3.
So the line rises 3 units for every 1 unit it moves to the right.
Sign and special cases
A positive gradient means the line slopes upward from left to right. A negative gradient means it slopes downward. A horizontal line has a gradient of 0 (no vertical change). For the equation y = mx + c, the number m in front of x is the gradient, so y = 2x + 3 has gradient 2.
| Line direction | Gradient |
| Uphill (left to right) | positive |
| Downhill (left to right) | negative |
| Flat / horizontal | zero |
The steeper the line, the larger the size of the gradient. Two points that are far apart give the same gradient as two points close together on the same line, because the ratio of vertical change to horizontal change stays constant along a straight line.
Remember
- Gradient = (y₂ − y₁) ÷ (x₂ − x₁).
- Keep the points in the same order on top and bottom.
- In y = mx + c, m is the gradient.