Average Rates of Change
Practise average rates of change with this free Foundation and Higher GCSE Maths worksheet from Virtus Academy. You'll work through calculating average rates of change, building confidence for your exam, and every question comes with worked solutions in the mark scheme. Suitable for AQA, Edexcel and OCR. This is the gradient of the chord between two points.
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Topic overview
An average rate of change measures how quickly one quantity changes relative to another over an interval. On a graph it is the gradient of the straight line joining two points on the curve, called a chord.
The calculation is the change in the vertical quantity divided by the change in the horizontal one, exactly as for the gradient of a line. On a distance-time graph this gives average speed; on a velocity-time graph it gives average acceleration.
Because it uses a straight chord across a curve, the result is an average rather than the rate at any single moment. That distinction matters, and questions often test whether you understand it.
Revision notes
Calculating the average rate
Take the two endpoints of the interval and find the gradient of the chord between them.
If a curve passes through \((1, 4)\) and \((5, 20)\), the average rate of change is \(\frac{20-4}{5-1} = 4\) per unit.
Interpreting in context
On a distance-time graph the rate is speed; on a velocity-time graph it is acceleration.
Always give units, such as metres per second, since the question is really about the real-world quantity rather than the number.
Average versus instantaneous
The chord gives an average across the whole interval, which may differ greatly from the rate at any particular instant.
A car averaging \(30\) mph over an hour may have been stationary at times and much faster at others.
Key points
- The average rate of change is the gradient of a chord.
- Divide the change in y by the change in x.
- On a distance-time graph it gives average speed.
- On a velocity-time graph it gives average acceleration.
- Always include units.
- An average may differ from the rate at any instant.
Worked examples
Example 1
A curve passes through \((2, 5)\) and \((6, 21)\). Find the average rate of change.
Working
Example 2
A distance-time graph shows \(150\)m covered between \(10\)s and \(40\)s. Find the average speed.
Working
Example 3
Explain why an average speed of \(20\) km/h does not mean travelling at \(20\) km/h throughout.
Working
Common mistakes
Using only one point.
A rate of change needs two points to form the chord.
Omitting units.
The answer describes a real quantity, so metres per second or similar is required.
Confusing average with instantaneous rate.
A chord gives an average; a tangent gives the rate at a single point.
Dividing the wrong way round.
The vertical change goes on top, exactly as for a gradient.
Exam tips
- Mark the two endpoints clearly on the graph.
- Draw the chord between them before calculating.
- Give the answer with the correct units for the context.
- State that the value is an average when the question asks for interpretation.
Key terms
- Chord
- A straight line joining two points on a curve.
- Rate of change
- How quickly one quantity changes relative to another.
- Average speed
- Total distance divided by total time.
- Interval
- The section between two values.
Related topics
Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.