When a graph is a straight line through the origin, the two variables are directly proportional: doubling one doubles the other. The gradient of the line tells you how much the y-variable changes for each unit of the x-variable. Gradient = change in y ÷ change in x.
Gradients often have a scientific meaning. On a distance–time graph the gradient is speed. For a spring, extension ÷ mass tells you how much the spring stretches per gram. Always include units with a gradient, for example mm per g or m/s.
Worked example
A spring stretches 40 mm with 200 g and 80 mm with 400 g. What is the gradient?
- Gradient = change in extension ÷ change in mass.
- (80 − 40) ÷ (400 − 200) = 40 ÷ 200.
Answer:0.2 mm per g.
Key words
- gradient
- the steepness of a line: change in y ÷ change in x
- directly proportional
- a straight line through the origin
- origin
- the point (0, 0)
- rate
- how fast something changes
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