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 and calculate its gradient. This skill is essential for interpreting distance-time graphs, velocity-time graphs, and real-world data involving flow rates, temperatures and populations. This guide covers the method with worked examples.

What Is a Rate of Change?

A rate of change measures how one quantity changes relative to another. The gradient of a graph gives you the rate of change of the y-variable with respect to the x-variable.

Key Formulas

For a straight line:

For a curve at a specific point, the rate of change equals the gradient of the tangent at that point.

Interpreting rate of change in context

• On a distance-time graph, the gradient gives speed.
• On a velocity-time graph, the gradient gives acceleration.
• On a volume-time graph, the gradient gives flow rate.
• A positive gradient means the quantity is increasing; a negative gradient means it is decreasing.
• A steeper tangent indicates a faster rate of change.

Step-by-Step Method

Drawing a tangent and finding the gradient of a curve

1. Plot or identify the point on the curve where you need the rate of change.
2. 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.
3. Choose two points on the tangent line that are far apart (to improve accuracy). Read their coordinates.
4. Calculate the gradient using (y₂ - y₁) / (x₂ - x₁).
5. Interpret the gradient in the context of the question (e.g., "the speed at t = 3 seconds is approximately 12 m/s").

Worked Example 1 — Finding Rate of Change from a Curve

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.

Suppose the tangent drawn at t = 5 passes through the points (3, 20) and (7, 60).

Working:

Gradient = (60 - 20) / (7 - 3) = 40 / 4 = 10 m/s

Answer: The speed at t = 5 seconds is approximately 10 m/s.

Worked Example 2 — Higher Level

Question: A tank is being filled with water. The volume V (litres) after t minutes is plotted on a curve. At t = 10 minutes, a tangent to the curve passes through (6, 30) and (14, 78). Find the rate at which water is entering the tank at t = 10 minutes.

Working:

Gradient = (78 - 30) / (14 - 6) = 48 / 8 = 6 litres per minute

Answer: At t = 10 minutes, water is entering the tank at 6 litres per minute.

Worked Example 3 — Exam Style

Question: The depth of water d (cm) in a container after t seconds is modelled by a curve. At t = 4, a tangent is drawn. It passes through (2, 18) and (6, 10). Find and interpret the rate of change at t = 4.

Working:

Gradient = (10 - 18) / (6 - 2) = -8 / 4 = -2 cm per second

Answer: The depth is decreasing at a rate of 2 cm per second at t = 4 seconds. The negative sign indicates the water level is falling.

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.

Practice Questions

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.

Practise rates of change questions with instant feedback free on GCSEMathsAI.

Related Topics

• Linear Graphs and Equation of a Line
• Quadratic Graphs
• Speed, Distance and Time
• Direct and Inverse Proportion

Summary

• The rate of change at a point on a curve equals the gradient of the tangent at that point.
• To find the gradient: draw a tangent, pick two well-spaced points on it, and calculate (y₂ - y₁) / (x₂ - x₁).
• On distance-time graphs, gradient = speed. On velocity-time graphs, gradient = acceleration.
• A positive gradient means the quantity is increasing; a negative gradient means it is decreasing.
• Always interpret the gradient in context with correct units.
• Use a ruler and sharp pencil for accurate tangent lines in exams.
• Choose points far apart on the tangent to minimise reading errors.