Key Idea: In modelling you pick a type of function that fits real data, find its numbers, and explain them. It always runs in four steps: choose, fit, interpret, comment.
| Behaviour | Model |
|---|---|
| Steady change | Linear |
| One peak | Quadratic |
| Multiplies each step | Exponential k·eʳᵗ |
| Levels off at a ceiling | Logistic L/(1+Ce⁻ᵏᵗ) |
| Repeats | Sinusoidal a·sin(b(t−c))+d |
| Different rule per interval | Piecewise |
The modelling cycle
- Choose — Match the behaviour of the data.
- Fit — Find the parameters by hand from conditions, or by GDC regression.
- Interpret — Explain every parameter in context: starting value, rate, ceiling, amplitude, period.
- Comment — Say where the model is reliable and where it breaks down.
A model is only finished when you have said what its numbers mean.
Mugs cost 6 each for the first 20, then 4 each. Find C(n) for n > 20. C(n) = 120 + 4(n − 20) = 4n + 40.
Using that model, what do 50 mugs cost? 4(50) + 40 = 240.
Sales rise fast, then level off. Which model? Logistic.
Daily temperature over a year. Which model? Sinusoidal, because it repeats.