Updated · By Pete Bromfield, former Head of Maths with IB examining experience

IA modelling, step by step · Sinusoidal

Sinusoidal model, step by step: y = a sin(b(x − c)) + d

AA SLAA HLAI SLAI HL

For anything that repeats — daylight, tides, temperatures, a Ferris wheel — find a, b, c and d by hand from the maximum, minimum and period, convert to the calculator's form a sin(bx + c) + d, then refine the fit by minimising the sum of squared residuals.

Guidance based on the checklist Pete Bromfield — a former Head of Maths with IB examining experience — uses with his own students. Not affiliated with or endorsed by the IB.

Example data, invented for this guide. The context is realistic, but the numbers were made up to show the method. Use your own measured or sourced data in your IA.

When to use it

The shape of the data

  • The data rise and fall repeatedly, with peaks the same height and troughs the same depth.
  • Peaks (and troughs) are equally spaced: a fixed period.
  • The curve is symmetric about a middle line.

The context

  • Anything driven by the Earth's rotation or orbit: daylight, temperature, tides.
  • Rotations: a Ferris wheel seat, a point on a bicycle wheel.
  • Seasonal sales or visitor numbers (check that the peaks really are equal).

Course fit: AA SL and HL use y = a sin(b(x + c)) + d; AI SL uses a sin(bx) + d and AI HL adds the horizontal shift and sine regression. Whatever your course, show the by-hand estimates and explain the refinement.

The example data

Hours of daylight on the 15th of each month for a city at about 52° N, read from a sunrise–sunset table (values rounded to 0.1 h and slightly perturbed for this example).

Example data: hours of daylight and month (1 = january)
ix: Month (1 = January)y: Hours of daylight (h)
117.9
229.4
3311.7
4414.1
5515.9
6616.8
7716.4
8814.9
9912.6
101010.2
11118.4
12127.5
Scatter graph of hours of daylight against month (1 = january) for the example data
Step 1 of any modelling IA: plot the data and describe the shape before fitting anything.

The method: estimate by hand, then refine

Steps 1–6 find each parameter from the graph, using the known period of 12 months. Steps 7–8 then let all four parameters vary to minimise the sum of squared residuals, which is what sine regression does.

Step 1 · Read the maximum and minimum

Highest point: ymax = 16.8 at x = 6. Lowest point: ymin = 7.5 at x = 12.

Step 2 · Amplitude a and principal axis d

a = (ymax − ymin)/2 = (16.8 − 7.5)/2 = 4.650

d = (ymax + ymin)/2 = (16.8 + 7.5)/2 = 12.15

The principal axis y = d is the middle of the oscillation; a is the distance from it to a peak.

Graph of the data with the fitted sinusoidal (by-hand estimate)
Estimate by hand, then refine: the fitted curves over the example data.

In the full worked example

  • The rest of the method: steps 3 to 8 and the residual plot
  • What the model tells you, in context
  • GDC key sequences: TI-84 Plus CE, TI-Nspire CX and Casio fx-CG50
  • Desmos regression syntax
  • How to write it up in your IA
  • Common mistakes
  • Is this good enough for Criterion E? The checklist

Evaluate this model

Questions to answer in your own words, about your own data. They are prompts, not sentences to copy.

  • Does the period match a known cycle in the context (a day, a year, a tide)?
  • Do the amplitude and the principal axis stay the same over time in reality, or do they drift?
  • Are the angles in radians, and is that stated?

Then work through the general questions on evaluating a model (extrapolation, the parameters, residuals, comparing models, outliers, assumptions and improvements) and the notes on accuracy and rounding. In the free Grapher, fit the model with ~ and open “What this model means” for the same prompts with your own numbers.

IA ideas that use a sinusoidal model

Explorations from our idea list whose mathematics includes this model. Each research question is a starting point to make your own.

How this earns marks

A sinusoidal model can earn marks under every criterion: the fit and its explanation under Criterion E, your judgement of it (residuals, the domain, what the parameters mean) under Criterion D, and the reasons you chose it under Criterion C. What examiners look for, criterion by criterion and the mistakes that cost marks are on the modelling guide.

Get feedback on your write-up Self-check your draft Model my data

Frequently asked questions

How do I convert a sin(b(x − c)) + d into a sin(bx + c) + d?

Expand the bracket: b(x − c) = bx − bc. So the calculator's constant is C = −bc, and the horizontal shift is c = −C/b.

Why does my calculator give a negative a for a sine model?

Different parameter sets can draw the same curve: changing the sign of a and shifting by half a period, or adding 2π to the constant, gives an identical graph. Convert to your preferred form and say so.

Next steps

Related models: Cubic and polynomials (and overfitting) · Piecewise models · Choosing and comparing models. Or model your own data, and see what the IA package adds.

Free: the IA checklist, criterion by criterion

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