Choose a model by matching the pattern of change and the context to the defining features of a function, then test whether the model fits the data and has a reasonable domain. In IB Mathematics: Applications and Interpretation, the model must be justified rather than selected only because it produces a high calculator value.
The Modelling Process
In SL 2.6, Modelling skills, follow the modelling process:
- Identify the variables and what they represent.
- Examine the data, graph, or contextual description.
- Choose a model whose mathematical behaviour matches the pattern.
- Determine its parameters using given points, initial conditions, simultaneous equations, or linear regression where appropriate.
- Test the model against the data and context.
- State a reasonable domain, use the model, and reflect on its limitations.
| Observed pattern | Appropriate model |
|---|---|
| Approximately constant absolute change | Linear, such as |
| One turning point or curved rise and fall | Quadratic, such as |
| Approximately constant percentage change | Exponential, such as |
| Repeating pattern with a fixed period | Sinusoidal, such as |
| More complex curve with up to two turning points | Cubic, such as |
For example, a population increasing by about each year suggests an exponential model because each value is approximately times the previous value. A model might therefore begin as , where is the initial population.
The model must also be valid in context. A negative predicted population is impossible, and a model fitted over five years may be unreliable when used to predict fifty years ahead. This is the distinction between safer interpolation and risky extrapolation.
Exam Technique
For a command term such as justify, name a specific feature connecting the data to the model: constant gradient, constant ratio, turning point, or periodicity. A common misconception is that the model with the closest numerical fit is automatically best; you must also discuss its domain, assumptions, and contextual reasonableness. This applies to both SL and HL questions, especially extended Paper 2 modelling tasks.
