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IB Math AI SL · Topic 2

Functions

The spine of Applications and Interpretation. Six model families, one repeating cycle examined everywhere: choose a model, justify it, find its parameters, state a reasonable domain, and interpret predictions back into context.

What the syllabus covers

Six assessed sections, SL 2.1 to 2.6 — less algebra than AA, far more interpretation.

SL 2.1Straight linesSlope–intercept, general and point–slope forms; gradient; parallel lines m₁ = m₂; perpendicular lines m₁ × m₂ = −1.
SL 2.2Functions, domain and rangeNotation such as f(x), v(t), C(n); the function as a mathematical model; the inverse as the undo operation, reflected in y = x.
SL 2.3Graphing skillsSketching from information given or a context; transferring graphs from the GDC screen to paper; graphing sums and differences with technology.
SL 2.4Key features of graphsMaxima and minima, intercepts, horizontal and vertical asymptotes; intersections of two curves found with technology.
SL 2.5The model familiesLinear mx + c; quadratic with axis, vertex and zeros; exponential growth/decay k·aˣ + c with its horizontal asymptote; direct/inverse variation; cubic; sinusoidal a·sin(bx) + d.
SL 2.6The modelling processChoose a model in context, fit its parameters, state a reasonable domain, justify the choice from data shape and curve properties, then read predictions back out.

How it is examined

Modelling chains are Paper 2's signature — one real-world scenario unfolding over five or six connected parts.

  • Real papers open with extended sinusoidal models — a Ferris wheel's height over time, tides in a harbour — where you must find the parameters before you may use them.

  • Marks are reserved for interpretation sentences: what the asymptote means, whether the domain is reasonable, why this model fits this data.

  • Parameter-hunting questions expect the working that produced them, not just the values.

  • The formula booklet gives exactly two things for this topic: the three forms of a straight line with the gradient formula, and the axis of symmetry x = −b/2a. Everything else — quadratics solved, exponentials fitted, sinusoids constructed — is examined as GDC fluency plus interpretation.

  • Command terms distinguish three kinds of graph question: 'sketch' wants shape and key features freehand, 'draw' wants an accurate curve on labelled axes, 'plot' wants points marked precisely — and examiners mark the difference.

  • Transferring a graph from the GDC screen to paper is an explicitly listed skill: shape, labelled axes and intercepts, not a copy of pixels.

A marked question, mark by mark

A sinusoidal model built from a context, marked the way the scheme would mark it.

Drill the full modelling cycle, marked

Generated exercises across all six SL 2.x model families, marked line by line — including the interpretation sentences that static question banks can't teach because their answers are already printed.

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New to the codes? M1, A1, R1 explained →

Sinusoidal models · SL 2.5 / 2.6

Seed 5C13

Water depth in a harbour is modelled by D(t) = a sin(bt) + c metres, with t in hours. The depth oscillates between 3 m and 9 m with a period of 12 h.

  1. (a)Find the values of a, b and c.
  2. (b)Interpret the meaning of c in this context.

Your work

a = (9 − 3)/2 = 3

c = (9 + 3)/2 = 6

b = 2π/12 = π/6

Marks awarded

  • M1Mark awarded. Amplitude taken as half the range and axis as the midpoint — correct use of the model's structure.
  • A1Mark awarded. a = 3, b = π/6, c = 6 all correct.
  • R1Mark not awarded. c reported as a number only — never linked to the mean water depth about which the tide oscillates.

Where marks are actually lost

The interpretation layer is where this topic is decided.

Choosing a model family from the scatter alone, without justifying it from the context or the data's behaviour.

CostAppropriateness is a marked judgement, not a formality.

Predicting far outside the observed data with no comment on the domain.

CostExtrapolation without a reasonable-domain statement loses the communication mark attached to it.

Reporting regression or model parameters without saying what each represents in context.

CostA parameter named correctly is a mark; a bare number is not.

Sketching from memory instead of transferring key features from the GDC screen.

CostShape, intercepts and labels are individually assessed — a generic curve earns none of them.