Rate of change describes how one quantity changes with respect to another. On a graph, the rate of change is found from the gradient. At Higher tier this includes average and instantaneous rates of change.
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Key facts to remember
1Rate of change = gradient of a graph = change in y ÷ change in x.
2On a distance-time graph, gradient = speed.
3On a velocity-time graph, gradient = acceleration.
4Average rate of change: gradient of the chord between two points.
5Instantaneous rate of change: gradient of the tangent at a point (drawn by hand at GCSE).
6A steeper gradient means a faster rate of change.
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Formulas
Gradient (rate of change)
gradient = (y₂ − y₁) / (x₂ − x₁)
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Worked examples
Example 1
A graph shows distance (km) against time (hours). Between t = 1 and t = 4, the distance goes from 20 km to 80 km. Find the average speed.
Working
Average rate of change = gradient of chord
Gradient = (80 − 20) / (4 − 1) = 60 / 3 = 20 km/h
Answer20 km/h
Example 2
On a curved graph, a tangent is drawn at t = 3. The tangent passes through (1, 10) and (5, 30). Find the instantaneous rate of change at t = 3.