Growth and decay problems (including compound growth and depreciation), interpreting the gradient of a graph as a rate of change, and estimating the gradient of a curve and the area under a graph (Higher tier).

What this dot point is asking

Edexcel expects you to model repeated growth and decay with a multiplier (compound growth, depreciation and population change), and at Higher tier to interpret the gradient of a graph as a rate of change and to estimate the gradient of a curve or the area under a graph. These ideas connect percentages, graphs and real-world rates, and they reward clear reasoning about what a graph's features mean.

Compound growth and decay

Compound growth and decay apply the same percentage change repeatedly, so a multiplier is raised to a power.

So a population of $8000$ growing at $3\%$ per year for $5$ years is $8000 \times 1.03^5 \approx 9274$. A machine worth $£20000$ depreciating at $12\%$ per year for $4$ years is $20000 \times 0.88^4 \approx £11\,994$. The crucial point, as with compound interest, is that the percentage applies to the changing amount each period, not the original.

Gradient as a rate of change

The gradient of a graph measures how fast one quantity changes with another, which is a rate.

On a distance-time graph, the gradient is the speed: a steeper line means faster motion, a horizontal line means stationary. On a speed-time graph, the gradient is the acceleration, and a negative gradient means deceleration. Reading the units of the axes tells you what the gradient represents, which is the heart of these interpretation questions.

Area under a graph

On a speed-time graph, the area between the line and the time axis gives the distance travelled, because distance is speed multiplied by time.

For a curve, the area is estimated by splitting the region into strips (usually trapeziums) and adding their areas. More strips give a closer estimate.

Estimating the gradient of a curve

When a graph is curved, the rate of change is different at every point. To estimate the gradient at a particular point, draw a tangent (a straight line just touching the curve there), then find the gradient of that tangent using two points on it. This gives, for example, the instantaneous speed from a curved distance-time graph. The estimate depends on how well the tangent is drawn, so questions accept a range of answers.

Reading real-world graphs

Many exam questions wrap these ideas in a context such as a journey, a filling tank or a cooling drink, and reward you for describing what the graph shows. A flat section of a distance-time graph means no movement; a steeper section means a faster speed. On a graph of volume against time for a tank filling up, a steeper line means a faster flow rate, and a horizontal line means the tank is full. The phrase "interpret the gradient" is asking you to translate a number into a sentence about the situation, with the correct units, so always state what the rate physically represents.

Try this

Q1. A flat costs $£150000$ and rises in value by $4\%$ per year. Work out its value after $2$ years. [3 marks]

• Cue. $150000 \times 1.04^2 = £162240$.

Q2. On a distance-time graph, a line goes from $(0, 0)$ to $(5, 100)$, with distance in metres. Work out the speed. [2 marks]

• Cue. Gradient $= \dfrac{100}{5} = 20\,\text{m/s}$.