Sheet № 234 · Higher only · AQA · Edexcel · OCR
Rates of Change –
Rates of change is a Higher tier topic in GCSE Maths that asks you to find and interpret how quickly a quantity is changing at a particular moment. While straight-line graphs have a constant gradient, curves have a gradient that varies from point to point. To find the rate of change at a specific point on a curve, you draw a tangent line
§Key definitions
Question:
The graph below shows the distance (in metres) travelled by a cyclist over time (in seconds). Estimate the speed of the cyclist at t = 5 seconds.
Answer:
The speed at t = 5 seconds is approximately 10 m/s.
Q1 (Higher):
A curve passes through the point (4, 25). The tangent at this point also passes through (2, 15). Estimate the gradient at x = 4.
Q2 (Higher):
On a distance-time graph, a tangent at t = 8 passes through (5, 40) and (11, 100). What is the speed at t = 8?
Q3 (Higher):
A volume-time curve shows the tangent at t = 6 passing through (3, 50) and (9, 20). Find and interpret the rate of change.
§Formulas to memorise
Rate of change = Gradient = Change in y / Change in x = (y₂ - y₁) / (x₂ - x₁)
Plot or identify the point — on the curve where you need the rate of change.
Draw a tangent line — at that point. The tangent should just touch the curve at the point and have the same direction as the curve there. Use a ruler.
Choose two points on the tangent line — that are far apart (to improve accuracy). Read their coordinates.
Calculate the gradient — using (y₂ - y₁) / (x₂ - x₁).
Interpret the gradient — in the context of the question (e.g., "the speed at t = 3 seconds is approximately 12 m/s").
Worked example
The graph below shows the distance (in metres) travelled by a cyclist over time (in seconds). Estimate the speed of the cyclist at t = 5 seconds. Suppose the tangent drawn at t = 5 passes through the points (3, 20) and (7, 60).
Working:
⚠ Common mistakes
- ✗Drawing a chord instead of a tangent. A chord connects two points on the curve and gives the average rate of change, not the instantaneous rate. A tangent touches the curve at one point only.
- ✗Choosing points too close together on the tangent. This magnifies reading errors from the graph. Pick points that are far apart on the tangent line.
- ✗Forgetting to interpret in context. The gradient is a number, but the question usually asks what it means. State the rate with correct units and explain whether the quantity is increasing or decreasing.
- ✗Ignoring negative gradients. A negative gradient means the quantity is decreasing. Do not drop the negative sign.
✦ Exam tips
- →Use a sharp pencil and a ruler when drawing tangents. Examiners accept answers within a reasonable tolerance, but a carelessly drawn tangent can lead to a significantly wrong gradient.
- →Label the two points you use on the tangent and show the gradient calculation. This earns method marks even if your tangent is slightly off.
- →If the question says "estimate," the examiner expects you to draw a tangent and read values from the graph. An approximate answer is acceptable.
- →Practice reading graph scales carefully. Misreading a value by one small square can change your gradient significantly.