AHL 2.9 (HL)—Further modelling functions

AHL models extend the SL families: half-life uses exponential decay; $a+b\ln x$ models logarithmic change; $a\sin(b(x-c))+d$ models cycles; $L/[1+Ce^{-kx}]$ models growth limited by carrying capacity $L$; piecewise functions use different rules on stated intervals.

For a half-life $h$, $Q(t)=Q_0(1/2)^{t/h}$. In a sinusoidal model, radians are assumed unless degrees are marked, the period is $2\pi/|b|$, and $c$ is a horizontal translation or phase shift. In a logistic model, $L$ is the horizontal asymptote and carrying capacity.

Example

If a quantity starts at 800 and has half-life 3 years, $Q(t)=800(1/2)^{t/3}$, so $Q(6)=200$. For a piecewise model, solve shared-boundary parameter conditions when continuity is required; the formal definition of continuity is not required.

A model family is chosen from mechanism and shape, not fit alone. Do not use degrees in a radian model, call $L$ the initial value of a logistic curve, or assume two piecewise rules join continuously without checking their boundary values.