Key Idea: Real-world relationships rarely look like straight lines. Topic 2.5 is about recognising which mathematical model fits a given situation — linear, quadratic, exponential, power, or sinusoidal — and understanding what the parameters of each model actually mean. Choosing the right model is a skill that requires looking at the shape of the data and the context of the problem.
📊 The five model types
| Model | Equation form | Key feature | Real-world example |
|---|---|---|---|
| Linear | y = mx + c | Constant rate of change — straight-line graph. m = rate, c = starting value. | Monthly phone bill with fixed charge + per minute cost |
| Quadratic | y = ax² + bx + c | U-shaped (a > 0) or arch (a < 0). Has a vertex (max or min). Non-constant rate. | Projectile height; profit vs price |
| Exponential | y = a·bˣ | b > 1 → growth. 0 < b < 1 → decay. Horizontal asymptote at y = 0 (or shifted). | Population growth; radioactive decay |
| Power/variation | y = a·xⁿ | n > 1: accelerates upward. n < 0: decreasing. Passes through or near origin. | Surface area vs radius; speed vs drag |
| Sinusoidal | y = a·sin(bx) + d | Repeating waves. a = amplitude, 2π/b = period, d = vertical shift. | Tides; temperature over months; Ferris wheel height |
🔢 Reading model parameters from context
Example: Exponential model: A population starts at 500 and grows by 8% per year. y = 500 × (1.08)ˣ — here a = 500 (starting value), b = 1.08 (growth factor). Sinusoidal model: Temperature varies between 10°C and 30°C with a 12-month period. Amplitude a = (30−10)/2 = 10. Vertical shift d = (30+10)/2 = 20. Period 12 → b = 2π/12 = π/6. y = 10 sin(πx/6) + 20
Recognising which model to use: - Constant difference → linear - Constant ratio → exponential - Symmetric, vertex-like shape → quadratic - Repeating, wave-like → sinusoidal - Power law (doubles when x quadruples, etc.) → power model For sinusoidal: amplitude = (max − min)/2. Period = time for one full cycle.
Paper 2 (GDC allowed): When asked to fit a model to data, use GDC regression. The GDC gives you the equation — read off the parameters and interpret them in context. Paper 1: You may be given a model equation and asked to interpret: what does the coefficient a mean? What does the exponent tell you? Always tie your answer to the real-world context in the question.
IB-style question [6 marks]
For each data set below, state which model — linear, quadratic, exponential, or sinusoidal — fits best, and give a brief reason. (a) A savings account that increases by 4% each year. (b) The height of a ball thrown straight up, recorded each second until it lands. (c) The number of hours of daylight recorded on the 1st of each month over two years. (d) The cost of a taxi that charges a fixed flag-fall plus a fixed amount per kilometre.
Step by step:
(a) A constant percentage change each step means a constant multiplier, so the model is exponential.
(b) The height rises to a single peak then falls symmetrically — one turning point — so the model is quadratic.
(c) Daylight rises and falls in a repeating yearly cycle, so the model is sinusoidal.
(d) A fixed amount plus a constant rate per kilometre is a constant rate of change, so the model is linear.
(a) Exponential — constant percentage change. (b) Quadratic — single symmetric peak. (c) Sinusoidal — repeating cycle. (d) Linear — constant rate of change.