AHL 2.9—HL modelling functions

At Higher Level, Math AI extends modelling beyond simple linear and quadratic functions into a rich toolkit of function types that can describe complex, real-world behaviour. This subtopic covers five key model families:

• Exponential models , rapid growth or decay (radioactive decay, compound interest)
• Natural logarithmic models , diminishing returns, perceived sensation
• Sinusoidal models , periodic/cyclical phenomena (tides, temperature, sound)
• Logistic models , growth with a carrying capacity (populations, disease spread)
• Piecewise models , behaviour that changes abruptly at defined thresholds

A critical HL skill is not just applying these models, but selecting the most appropriate one given data or context, and fitting parameters using either algebra or a GDC.

Exponential Model: A function of the form $f(x)=a\cdot b^x$ (or equivalently $f(x)=a\cdot e^{kx}$) where the rate of change is proportional to the current value. If $k>0$ the model describes growth; if $k<0$ it describes decay.

The decay form most commonly used in physics and pharmacology is:

$N(t)=N_0\cdot e^{-\lambda t}$

where:

• $N(t)$ = quantity remaining at time $t$
• $N_0$ = initial quantity
• $\lambda$ = decay constant (always positive)

Half-Life: The time $T_{1/2}$ required for a quantity to reduce to exactly half its current value. It is related to the decay constant by:
$T_{1/2}=\frac{\ln 2}{\lambda }$

Deriving the half-life formula: Set $N(t)=\frac{N_0}{2}$:
$\frac{N_0}{2}=N_0\cdot e^{-\lambda T_{1/2}}$
$\frac{1}{2}=e^{-\lambda T_{1/2}}$
$\ln \left(\frac{1}{2}\right)=-\lambda T_{1/2}$
$T_{1/2}=\frac{\ln 2}{\lambda }$