Preview
IB DP Mathematics: Applications and Interpretation | SL & HL | Unit 2: Functions
📋 Learning Intentions
- Identify the appropriate function type (linear, quadratic, exponential, logarithmic, sinusoidal, logistic) to model a given real-world data set.
- Fit a mathematical model to data using GDC regression tools and interpret the parameters in context.
- Evaluate the suitability of a model using the coefficient of determination \( r^2 \), residuals, and contextual reasoning.
- Use a fitted model to make predictions (interpolation and extrapolation) and comment on their reliability.
- Justify the choice of model by comparing goodness-of-fit statistics and the behaviour of the function against the data.
- Interpret the parameters of a model (gradient, initial value, growth rate, amplitude, period) in the context of the problem.
Real-World Introduction
A marine biologist records the population of a fish species in a lake every two years. The counts are: 120, 310, 780, 1950, 4800. No formula was given — the biologist must decide which mathematical function best describes this growth, fit it to the data, and use it to predict future populations. This is mathematical modelling in practice.
Epidemiologists used logistic growth models to predict the spread of COVID-19. Climate scientists use exponential decay models for radioactive tracers in ocean currents. Engineers use sinusoidal models for tidal heights. In every case, the process is the same: collect data, choose a model type, fit it, test it, and use it — always with an awareness of its limitations.
🌍 Why This Matters Mathematical modelling is the bridge between raw data and actionable decisions. The IB Math AI course treats modelling as a core skill: every regression, every distribution, every optimisation problem is an instance of the same four-stage cycle — develop, fit, test, use.
Core Concepts & Definitions
Before fitting any equation to data, a decision must be made: what family of functions is appropriate? This decision is guided by the shape of a scatter plot, the context of the problem, and knowledge of how different function types behave. Choosing the wrong model type — even with a perfect fit — can lead to nonsensical predictions.
🔑 Key Definition — Mathematical Model A mathematical model is a function or equation that describes the relationship between variables in a real-world situation. In IB Math AI, models are always chosen from a defined set of function families and are fitted to data using regression techniques on the GDC.
A model is not the same as reality — it is a simplification. The goal is to find a function that captures the essential behaviour of the data well enough to be useful for prediction and interpretation. Every model has limitations, and part of the modelling skill is knowing when a model is reliable and when it is not.
The Four-Stage Modelling Cycle
IB Math AI structures mathematical modelling as a four-stage cycle. Each stage has specific mathematical techniques and exam expectations attached to it.
- Stage 1 — Develop: Identify the type of model from the context and scatter plot. State the general form of the function.
- Stage 2 — Fit: Use GDC regression to find the specific parameters (e.g. values of \(a\), \(b\), \(r^2\)).
- Stage 3 — Test: Evaluate the