This HL mini-lesson deepens modelling: the SL families plus HL extensions — logistic models (limited growth), piecewise and natural log models, richer sinusoidal modelling, and careful use of regression and residuals to compare competing models.
AI HL flavour: you must select, fit and critique models — including when growth is capped (logistic) rather than unlimited (exponential).
Work through each screen, answer the questions as you go (some are wordy, some are calculations) and collect ⭐ stars. Every number on the calculation screens has been re-derived and checked. Press Start when you are ready.
Least-squares regression gives y = a + bx with correlation r and r². Judge fit with r² and the pattern of residuals.
Data gives y = 44.9 + 6.9x, r = 0.995, r² = 0.989.
Predict x = 6: y = 44.9 + 6.9×6 = 86.3
A high r² is not enough — check residuals show no curve or pattern, or a non-linear model is better.
Real populations cannot grow forever. The logistic model levels off at a carrying capacity L:
L = 1000, C = 9, k = 0.5, t = 4.
f(4) = 1000 ⁄ (1 + 9 e−2) = 1000 ⁄ (1 + 1.218) = 450.85
A logarithmic model f(x) = a + b ln x rises quickly then flattens — good for diminishing returns.
a = 2, b = 5, x = 10.
f(10) = 2 + 5 ln 10 = 2 + 5 × 2.3026 = 13.51
f(t) = k·at grows by a fixed percentage each period. A colony of 100 at +8%/yr is P = 100 × 1.08t, giving 216 after 10 years.
f(t) = a sin(b(t − c)) + d models cycles: a = amplitude, period = 2π⁄b, d = midline, c = horizontal shift. Used for tides, daylight and temperature.
f(t) = 8 sin(0.5 t) + 15.
At t = π: f(π) = 8 sin(0.5π) + 15 = 8 × 1 + 15 = 23
Tap a described behaviour, then the model type.
Tap an item on the left, then its match on the right.
Some situations change rule at a threshold — e.g. a tax band or a tiered tariff. A piecewise function uses different formulas on different intervals.
Check the pieces agree (or note the jump) at the boundary values.
Regression: y = a + bx; judge with r², residuals
Exponential: k·a^t — unlimited constant-% growth
Logistic: L ⁄ (1 + Ce^(−kt)) — growth capped at L
Logarithmic: a + b ln x — diminishing returns
Sinusoidal: a sin(b(t−c)) + d — amplitude a, period 2π⁄b
Skill: select, fit and critique the best model in context
You have worked through the whole topic. Press Finish to see your score.
You have covered HL Functions & Modelling for AI. 🎉
Your stars: 0 / 0
Next: test yourself in the Evaluate stage, then lock it in with Verify.