Find the gradient of the line through (−1, 8) and (3, −4).
- Change in y = −4 − 8 = −12.
- Change in x = 3 − (−1) = 4.
- Gradient = −12 ÷ 4 = −3.
Answer: Gradient = −3
Gradient of linear graphs
Maths · Coordinate geometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
The gradient of a straight line measures how steep it is. It is the change in y divided by the change in x as you move along the line, often called rise over run.
A line that goes up from left to right has a positive gradient. A line that goes down from left to right has a negative gradient. A horizontal line has gradient 0, and a vertical line has no defined gradient.
The gradient is the same everywhere on a straight line, so any two points on it give the same answer. In the form y = mx + c, the number m multiplying x is the gradient.
Rules to remember
- gradient = change in y ÷ change in x (rise ÷ run)
- For points (x1, y1) and (x2, y2): m = (y2 − y1)/(x2 − x1)
- In y = mx + c the gradient is m; rearrange to this form first if needed
- ax + by = c has gradient −a/b
- Horizontal line (y = b): gradient 0; vertical line (x = a): gradient undefined
- Three points lie on one straight line if the gradient between each pair is the same
Common mistake
Subtracting in a different order on the top and the bottom, or putting the x-change on top. Start from the same point in both subtractions and take care with double negatives such as 3 − (−2) = 5.
Worked example
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