SL 2.5—Modelling with functions

A mathematical model uses a function to represent a real relationship. The model is useful only over a stated context and with assumptions about variables, units and measurement.

Choose a form from behaviour: constant change suggests linear, repeated proportional change suggests exponential, and a single peak may suggest a quadratic. Fit parameters from data, then test residuals or predictions and state the domain where the model is credible.

A population rising by roughly 4% per year may be modelled by P(t)=P₀(1.04)^t. The model predicts a value, not a biological law; resource limits can make the extrapolation fail.

A close fit does not validate an unlimited extrapolation. Explain why the chosen form matches the observed mechanism and where the assumption breaks.

Syllabus model map: linear and piecewise-linear models represent constant rates or changing tariffs; quadratics use roots, vertex and symmetry; exponentials use a repeated factor and horizontal asymptote; $f(x)=ax^n$ models direct or inverse variation and may have a vertical asymptote when $n<0$; cubics allow two turns; sinusoids $a\sin(bx)+d$ or $a\cos(bx)+d$ have amplitude $|a|$, period $360^\circ/|b|$ and principal axis $y=d$. Use technology for roots and parameters, then interpret them in context.