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Ratio & Proportion · Higher

Rates of change

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

  1. Average rate of change = gradient of chord
  2. 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.

Working

  1. Gradient of tangent = (30 − 10) / (5 − 1) = 20 / 4 = 5
AnswerInstantaneous rate of change = 5 units per unit time
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Common mistakes

✗Calculating the gradient using horizontal ÷ vertical instead of vertical ÷ horizontal.
✗Confusing average rate of change (chord) with instantaneous rate of change (tangent).
✗Not including units in the rate of change (e.g. km/h, m/s).
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Exam tips

✓For average rate of change, calculate the gradient between the two given points.
✓For instantaneous rate of change, draw a tangent at the point and calculate its gradient using two points on the tangent (not the curve).

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