Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Which function best describes the arch of a famous bridge?

AA SLAI SL Accessible Also in: Geometry & Voronoi, Art & music

Research question

Is the arch of [a chosen bridge] better modelled by a quadratic, a cosine curve or a semi-ellipse, judged by the sum of squared residuals?

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Why it makes a good exploration

Arches are chosen for strength, not looks, so the “right” curve depends on how the bridge carries its load. Comparing three families of curves on a structure you have visited gives a clear, personal modelling question with a satisfying answer.

The mathematics you'll need

Quadratic, sinusoidal and elliptical models
Transformations of functions to fit a model through known points
Residuals and sum of squared residuals
Symmetry and domain restrictions

Modelling the data, step by step

This idea usually needs sinusoidal or quadratic models. See it worked step by step, with a criterion tip at every step: Sinusoidal y = a sin(b(x − c)) + d · Quadratic: three points, completing the square, regression · Choosing and comparing models.

Model your own data Paste it from Desmos, GeoGebra or a spreadsheet and get the same play-by-play with your numbers. New to modelling? Start with the modelling workflow. Writing it up? The IA modelling planner comments on each paragraph as you draft — it never writes it for you.

Where the data comes from

Use a front-on photograph with a known dimension (span or height from the bridge's published details) and read 15+ points in GeoGebra.

GeoGebra — Free geometry and graphing software — Voronoi diagrams, loci and regression built in.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Choose a bridge you have a connection to; find its span and rise.
Set axes and extract points from the photograph.
Fit each family through the key points, then improve with least squares.
Compare residuals; discuss which shape the engineers probably intended and why.

Pitfalls that cost marks

Photos taken off-centre make the arch asymmetric.
Fitting only through three points and calling it a model — use all your data to judge fit.
Forgetting to convert pixels to metres consistently.

Showing personal engagement

Visit and photograph the bridge yourself.
Look up how the bridge was designed and test whether your best-fitting curve agrees.
Compare two bridges of different eras or materials.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Quadratic, sinusoidal and elliptical models; Transformations of functions to fit a model through known points
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Quadratic, sinusoidal and elliptical models; Transformations of functions to fit a model through known points
AI HL	Not a natural fit	The mathematics is mainly from the AA course; an AI HL version would need modelling with technology, statistics or networks at HL level.

Level: Accessible. A good first extended piece of maths, with room to go deeper. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: photos taken off-centre make the arch asymmetric; and fitting only through three points and calling it a model — use all your data to judge fit. Say how each one affects your answer.

Use of mathematics (E)

SL: quadratic, sinusoidal and elliptical models; transformations of functions to fit a model through known points; residuals and sum of squared residuals; symmetry and domain restrictions, each explained in your own words and carried out correctly.

How modelling explorations reach the top bands in general: Modelling with functions IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Extend by computing the area under the arch (for the opening) by integration or the trapezoidal rule, and compare the three models.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL modelling exploration adds.

See a complete IA, marked

When is the sea warm enough for our swimming club? A sinusoidal model of sea temperature (AI SL) is a complete modelling exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AA SL and AI SL. We rate it Accessible: a good first extended piece of maths, with room to go deeper.

What mathematics does this IA use?

Quadratic, sinusoidal and elliptical models, transformations of functions to fit a model through known points, residuals and sum of squared residuals and symmetry and domain restrictions.

Where can I get data for this IA?

Use a front-on photograph with a known dimension (span or height from the bridge's published details) and read 15+ points in GeoGebra. Checked sources and tools: GeoGebra and Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “Which function best describes the arch of a famous bridge?” filled in: the question, the maths and the data sources.

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