IB Maths AI modelling: choosing, fitting and judging a model

Match the behaviour to the model

Worked example 1: an exponential model from two data points

A colony has 50 bacteria at t = 0 hours and 72 at t = 2 hours. Model the population as P = a × b^t.

1. At t = 0, P = a × b^0 = a, so a = 50.
2. At t = 2, 72 = 50b^2, so b^2 = 1.44 and b = 1.2 (b must be positive).
3. Model: P = 50 × 1.2^t. Interpret b: the population grows by 20% each hour.
4. Predict t = 5: 50 × 1.2^5 = 50 × 2.48832 = 124 bacteria (to the nearest whole bacterium).
5. Check: P(2) = 50 × 1.44 = 72, which matches the data.
6. Judge the model: exponential growth cannot continue forever because food and space run out, so long-range predictions are unreliable. This is exactly where an HL logistic model improves on it.

Worked example 2: a quadratic model and its domain

A ball's height in metres is modelled by h(t) = -5t^2 + 20t + 1.5, where t is time in seconds.

1. Maximum height: the axis of symmetry is t = -20/(2 × -5) = 2. Then h(2) = -20 + 40 + 1.5 = 21.5 m.
2. When does the ball land? Solve -5t^2 + 20t + 1.5 = 0 with the GDC equation solver: t = 4.07 s or t = -0.0736 s.
3. Reject the negative root because time starts at 0. Check with the quadratic formula: t = (20 + sqrt(430))/10 = 4.07.
4. State the domain: 0 ≤ t ≤ 4.07. Outside this the model gives negative heights, which are meaningless.

Worked example 3: a sinusoidal model

The depth of water in a harbour varies between 3 m and 9 m. High tide is at t = 4 hours and the cycle repeats every 12 hours. Find a model h(t) = a sin(b(t - c)) + d.

1. Amplitude a = (9 - 3)/2 = 3. Principal axis d = (9 + 3)/2 = 6.
2. Period 12, so b = 2π/12 = π/6 (about 0.524). Work in radians.
3. The sine curve reaches its maximum a quarter-period after its start, and a quarter of 12 is 3. The maximum is at t = 4, so c = 4 - 3 = 1.
4. Model: h(t) = 3 sin((π/6)(t - 1)) + 6.
5. Check: h(4) = 3 sin(π/2) + 6 = 9, and h(10) = 3 sin(3π/2) + 6 = 3. Both match.

Fitting models to data with the GDC

When a question gives a table of data, the calculator's regression menu fits linear, quadratic, cubic, exponential, power and sinusoidal models. Enter the data, choose the family, write the model with parameters to 3 significant figures, and then use the model, not the raw data, for predictions. If the question asks which model fits best, compare the fit (for example the coefficient of determination R^2, if your level covers it) and also the shape: a model that fits the data points well but predicts something impossible outside them is not the best model.

At HL, data that looks exponential or like a power function can be linearised: taking logarithms turns y = ka^x into a straight line in ln y against x, and y = ax^n into a straight line in ln y against ln x. HL students also model with differential equations, including slope fields, Euler's method and coupled systems, and HL Paper 3 often builds a full model across one long question.

Common mistakes

• Using degrees for sinusoidal models. Set the calculator to radians unless the question says otherwise.
• Leaving out the domain, or failing to reject a root that has no meaning in context (negative time, negative lengths).
• Rounding parameters to 1 or 2 significant figures and then using the rounded model, which drifts from the expected answer.
• Interpreting a parameter vaguely. 'b is the growth' earns less than 'the population grows by 20% per hour'.
• Extrapolating far beyond the data without comment.

Exam technique, and how one-to-one lessons help

Modelling questions usually run in the same order: set up, find parameters, predict, interpret, criticise. The last two parts are short but carry marks that students give away by answering with numbers only. Practise finishing every modelling question with one sentence on what the answer means and one on a limitation of the model.

In lessons, a tutor who teaches AI gives the student a real situation and asks them to choose the model before any calculation, then to defend the choice. That habit, practised on the shared whiteboard with the student's own GDC, is what makes unfamiliar Paper 2 contexts and HL Paper 3 feel manageable. It also feeds directly into the exploration, where a well-justified model is often the core of the mathematics.

Self-check

• A car worth 30000 loses 15% of its value each year. Write a model and find its value after 4 years. (Answer: V = 30000 × 0.85^t, 15700 to 3 s.f.)
• Find the vertex of h(t) = -4.9t^2 + 14.7t. (Answer: t = 1.5, h = 11.025)
• A sinusoidal model has maximum 20 and minimum 8. Find a and d. (Answer: a = 6, d = 14)
• Why might a linear model for a child's height against age fail at age 30?