Key Idea: Real-world relationships rarely look like straight lines. Topic 2.5 is about recognising which mathematical model fits a given situation โ linear, quadratic, exponential, power, or sinusoidal โ 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.
๐ The five model types
๐ข Reading model parameters from context
Example: Exponential model: A population starts at 500 and grows by 8% per year. y = 500 ร (1.08)หฃ โ here a = 500 (starting value), b = 1.08 (growth factor). Sinusoidal model: Temperature varies between 10ยฐC and 30ยฐC with a 12-month period. Amplitude a = (30โ10)/2 = 10. Vertical shift d = (30+10)/2 = 20. Period 12 โ b = 2ฯ/12 = ฯ/6. y = 10 sin(ฯx/6) + 20
Recognising which model to use: - Constant difference โ linear - Constant ratio โ exponential - Symmetric, vertex-like shape โ quadratic - Repeating, wave-like โ sinusoidal - Power law (doubles when x quadruples, etc.) โ power model For sinusoidal: amplitude = (max โ 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 โ 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.