Rate of change measures how quickly one quantity changes relative to another. On a graph, the average rate of change between two points is the gradient of the chord joining them, whilst the instantaneous rate of change at a single point is the gradient of the tangent there.
What is the average rate of change?
The average rate of change between two points on a graph is the gradient of the straight line (chord) connecting those two points.
$$\text{Average rate of change} = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$$
This is the same formula as the gradient of a straight line between two coordinates. It tells you: on average, how much does y change for each unit increase in x over that interval?
Worked example: A car's distance (km) from home at various times is given in the table.
| Time (hours) | Distance (km) |
|---|---|
| 0 | 0 |
| 1 | 40 |
| 2 | 70 |
| 3 | 120 |
Average speed (= average rate of change of distance) from t = 1 to t = 3:
$$\frac{120 - 40}{3 - 1} = \frac{80}{2} = \textbf{40 km/h}$$
Note: this does not mean the car travelled at exactly 40 km/h throughout that period — it is the average over the two-hour interval.
What is the instantaneous rate of change?
The instantaneous rate of change at a single point is the rate of change at that exact moment — not averaged over an interval. On a graph, it is the gradient of the tangent to the curve at that point.
A tangent is a straight line that just touches the curve at one point, without crossing it. Its gradient gives the rate of change at that instant.
Example in context: On a distance-time graph, the gradient of the tangent at t = 2 gives the exact speed at t = 2 hours, whereas the chord gradient gives the average speed over an interval.
How do you draw and measure a tangent?
To find the instantaneous rate of change at a point P on a curve:
- Place your ruler so it touches the curve at point P only, with the curve on the same side of the ruler on both sides of P.
- Draw the tangent line across the grid.
- Pick two clear grid points on the tangent line (not on the curve itself).
- Calculate the gradient: rise ÷ run.
Worked example: A tangent drawn at t = 2 on the distance graph passes through (0, 10) and (4, 90).
$$\text{Gradient} = \frac{90 - 10}{4 - 0} = \frac{80}{4} = \textbf{20 km/h}$$
So the instantaneous speed at t = 2 hours is approximately 20 km/h. The word "approximately" matters — a hand-drawn tangent introduces a small measurement error.
How do the two rates compare on a velocity-time graph?
| Context | Average rate of change | Instantaneous rate of change |
|---|---|---|
| Distance-time graph | Average speed over interval | Speed at a specific moment |
| Velocity-time graph | Average acceleration | Acceleration at a specific moment |
| Temperature-time graph | Average rate of temperature rise | Rate of temperature rise at one instant |
On a velocity-time graph, the gradient represents acceleration. The gradient of the chord gives average acceleration; the gradient of the tangent gives instantaneous acceleration.
How do you interpret positive, negative and zero rates of change?
- Positive gradient: y is increasing as x increases. In a distance-time graph, the object is moving away from the starting point.
- Negative gradient: y is decreasing as x increases. In a distance-time graph, the object is moving back towards the start.
- Zero gradient (horizontal tangent): y is not changing at that instant. In a distance-time graph, the object is stationary.
A curve that becomes steeper has an increasing rate of change; a curve that flattens has a decreasing rate of change. These observations do not require any calculation — they are visual interpretations.
What mistakes should you avoid?
- Using the chord instead of the tangent for the instantaneous rate. The chord gives the average over an interval, not the instantaneous value at a point. Always draw a tangent for instantaneous rate.
- Reading the gradient backwards. Gradient = rise/run. If you read run/rise you get the reciprocal — a different (and wrong) value.
- Choosing gradient points on the curve rather than on the tangent. Once you have drawn the tangent, your gradient calculation must use two points on the tangent line.
- Forgetting units. The rate of change inherits units from both axes: if y is in km and x in hours, the rate of change is in km/h.
Frequently asked questions
Is the gradient of the tangent the same as the derivative in calculus?
Yes. At A-level, differentiation gives you a formula for the gradient of the tangent at any point on a curve — in other words, a formula for the instantaneous rate of change. At GCSE, you estimate it graphically by drawing the tangent by hand. The conceptual idea is the same; the method differs.
How accurate is a hand-drawn tangent?
A hand-drawn tangent is an approximation. On a GCSE exam, the mark scheme usually allows a range of acceptable answers — typically ±10–20% of the expected value, depending on the question. To minimise error: use a sharp pencil, extend the tangent line as far as possible across the graph, and choose gradient-calculation points as far apart as the grid allows.
When would you use the average rate of change rather than the instantaneous?
Average rate of change is more useful when you want an overall summary — e.g. "the average speed for the whole journey". Instantaneous rate of change is more useful when you need to know what is happening at a specific moment — e.g. "how fast was the car accelerating at exactly 3 seconds?" Many real-world applications (speed cameras, medical monitoring) measure instantaneous rates.
Can rate of change be applied to non-physical contexts?
Absolutely. The gradient of a graph showing population against time gives population growth rate. The gradient of a costs-against-output graph gives marginal cost. The gradient of a savings-against-time graph gives the rate of saving. Anywhere two quantities are plotted, the gradient measures the rate at which one changes relative to the other. GCSE exam questions often set this in unfamiliar contexts to test understanding rather than recall.
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