Overview of 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.

Exam Tip

When you see a modelling question, always ask: Is the quantity growing/decaying without bound? Levelling off? Oscillating? Changing rule at a threshold? Your answer points directly to the model family.

Exponential Models and Half-Life

Exponential Model

A function of the form $f (x) = a \cdot b^{x}$ (or equivalently $f (x) = a \cdot e^{k x}$ ) 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:

$N (t) = N_{0} \cdot e^{- λ t}$

where:

$N (t)$ = quantity remaining at time $t$
$N_{0}$ = initial quantity
$λ$ = 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:
$T_{1/2} = \frac{l n 2}{λ}$

Deriving the half-life formula: Set $N (t) = \frac{N _{0}}{2}$ :
$\frac{N _{0}}{2} = N_{0} \cdot e^{- λ T_{1/2}}$
$\frac{1}{2} = e^{- λ T_{1/2}}$
$ln (\frac{1}{2}) = - λ T_{1/2}$
$T_{1/2} = \frac{l n 2}{λ}$

Example

Radioactive decay problem

A sample contains 100 g of a radioactive substance with a half-life of 5 hours. Find the amount remaining after 12 hours.

Step 1: Find $λ$ .
$λ = \frac{l n 2}{T _{1/2}} = \frac{l n 2}{5} \approx 0.1386 h^{- 1}$

Step 2: Write the model.
$N (t) = 100 \cdot e^{- (l n 2/5) t}$

Step 3: Evaluate at $t = 12$ .
$N (12) = 100 \cdot e^{- (l n 2/5) (12)} = 100 \cdot 2^{- 12/5} \approx 29.3 g$

Interpretation: After 12 hours (2.4 half-lives), just under 30% of the original sample remains.

Common Mistake

A common error is writing $λ = T_{1/2} / ln 2$ instead of $λ = ln 2/ T_{1/2}$ . Always check: a larger $λ$ should mean faster decay (shorter half-life), so $λ$ and $T_{1/2}$ must be inversely proportional.

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AHL 2.9—HL modelling functions

Overview of HL Modelling Functions

Exponential Models and Half-Life

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