Modelling with functions — piecewise, exponential, logistic (IB Math AI HL)

Key ideas

• Logistic model: P(t) = L ÷ (1 + C e^−kt). L is the carrying capacity (the limit), and growth is fastest when P = L/2.
• Exponential models grow without limit; a logistic model looks exponential at first and then levels off.
• A power model y = axᵇ becomes a straight line when you plot ln y against ln x.
• Compare models with the coefficient of determination R² and with common sense: a model that predicts a negative population is not suitable, whatever its R².
• State the domain where the model applies, and say why extrapolating far beyond the data is unreliable.

Worked example

A population of fish in a lake is modelled by P(t) = 1000 ÷ (1 + 9e^−0.4t), where t is in years. Find the initial population and the long-term population, and the time when the population reaches 500.

1. Initial: P(0) = 1000 ÷ (1 + 9) = 100 fish.
2. Long term: as t → ∞, e^−0.4t → 0, so P → 1000. This is the carrying capacity.
3. P = 500: 1 + 9e^−0.4t = 2, so e^−0.4t = 1/9.
4. −0.4t = ln(1/9), so t = ln 9 ÷ 0.4 ≈ 5.49 years.
5. 500 is half the carrying capacity, so this is also when the population grows fastest.

Answer: Initially 100 fish; long term 1000 fish; 500 fish after about 5.49 years.

Exam-style HL AI questions on this topic

• Advanced Modelling questions — 37 questions with mark schemes, plus free worked examples
• All HL AI Functions questions by topic

Do it on your GDC

A GDC is allowed on every AI paper. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Graphs: intersections, zeros, max and min on your calculator

See it on a graph

• Graph of the logistic curve
• Curve fitting
• Open the free Grapher

Maths IA ideas that use modelling with functions

• Is a hanging chain a parabola? Catenary versus quadratic models
• Modelling the Keeling curve: trend plus seasons
• Modelling bacterial growth in the biology lab
• Exponential or logistic? Modelling my country's population
• All HL AI IA ideas

Revise Modelling with functions for the exam

• HL AI revision notes — every unit summarised, with the key results to learn
• HL AI past papers — specimen papers and timed practice papers
• Formulas for Functions — on the one-page HL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: If every mathematical model leaves something out, how can models produce knowledge?

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S24 · Problem S88 · Problem S23 · Problem S87. All 22 →