HL modelling — choosing & fitting a model
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Flip to reveal answersHow do you choose which function family to model a scenario with?
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Question
How do you choose which function family to model a scenario with?
Answer
Match the behaviour: steady change → linear; one peak → quadratic; constant multiplier each step → exponential; fast-then-levels-off at a ceiling → logistic; repeats/oscillates → sinusoidal; different rule on each interval → piecewise.
Question
What is the key sign that a model should be logistic rather than exponential?
Answer
It grows fast at first but LEVELS OFF at a finite ceiling (carrying capacity). A plain exponential never flattens.
Question
In y = L/(1 + C·e^{−kt}), what does L represent and how do you find it?
Answer
L is the carrying capacity (long-run ceiling): as t → ∞, e^{−kt} → 0 so y → L.
Question
In y = L/(1 + C·e^{−kt}), what is the initial value (t = 0)?
Answer
Put t = 0: e⁰ = 1, so y(0) = L/(1 + C).
Question
In N = k·e^{rt}, what do k and r mean?
Answer
k is the starting value (at t = 0); r is the continuous growth (r > 0) or decay (r < 0) rate.
Question
In a sinusoidal model a·sin(b(t − c)) + d, what is each parameter?
Answer
a = amplitude (half the swing), d = midline (average level), period = 2π/b, c = horizontal shift.
Question
How do you write a bulk-discount cost as a piecewise function?
Answer
One linear rule up to the threshold, then a second linear rule (fixed cost so far + new lower rate × extra units) beyond it.
Question
After fitting a model, what should you always do for the marks?
Answer
Interpret each parameter in context AND comment on validity (where the model is reliable vs where extrapolation is unsafe).
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HL modelling (HL only)
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