Model exponential growth and decay over repeated periods; and interpret the gradient of a graph as a rate of change and the area under a graph in real-life contexts (Higher tier).

What this dot point is asking

OCR references R15 and R16 cover growth and decay over repeated periods (exponential change) and, at Higher tier, interpreting the gradient of a graph as a rate of change and the area under a graph in real-life contexts. This content ties the multiplier idea from percentages to graphs and to compound measures. It appears mainly at Higher tier and on the calculator paper, and gradient-as-rate questions are a reliable multi-mark feature that tests AO2 interpretation.

Exponential growth and decay

Growth and decay multiply by a fixed factor each period.

So a population of $20000$ growing at $8\%$ a year reaches $20000 \times 1.08^5 \approx 29387$ after $5$ years, and a £$12000$ car depreciating at $20\%$ a year falls to $12000 \times 0.8^4 =$ £$4915.20$ after $4$ years. Growth and decay are exponential, not linear, which is why a single percentage cannot just be multiplied by the number of years. Contexts include savings, populations, radioactive decay and depreciation.

Gradient as a rate of change

The gradient of a real-life graph measures how fast one quantity changes with another.

So a distance-time graph rising from $20$ m to $80$ m between $4$ s and $10$ s has gradient $\dfrac{80 - 20}{10 - 4} = 10$ m/s, the speed. For a curved graph, the rate of change at a point is the gradient of the tangent at that point, which you estimate by drawing the tangent and finding its gradient. A negative gradient on a distance-time graph means returning towards the start.

A flat (horizontal) section of a distance-time graph means the object is stationary, because the distance is not changing. On a speed-time graph, a flat section means constant speed (zero acceleration), and a downward slope means the object is slowing down (deceleration). Being able to translate the shape of a graph into a word description of the motion, and the reverse, is a frequent OCR question that targets AO2 communication, so practise describing each segment in plain English.

Area under a graph

The area under certain graphs has a physical meaning.

Why this matters

Growth and decay model the real financial and natural world, and OCR sets them alongside compound interest because they share the multiplier idea. Gradient-as-rate and area-under-a-graph are the GCSE foundations of calculus, and they appear in physics-style questions where a graph must be read for speed, acceleration or distance. Stating what the gradient or area represents, in words, is exactly the AO2 communication OCR rewards.