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

How fast do cars lose value? Modelling depreciation from listings

AI SLAI HLAA SL Accessible Also in: Modelling, Statistics

Research question

Is the price of [a car model] better modelled as a linear or exponential function of age, what annual depreciation rate does the exponential model imply, and does mileage explain the residuals?

Make it yours, then agree it with your teacher.

Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.

Why it makes a good exploration

Depreciation is often quoted as a fixed percentage per year. Collecting real listings and testing that claim is simple to do, statistically rich and useful.

The mathematics you'll need

Linear and exponential regression
Logarithmic transformation
Interpreting the fitted rate as percentage depreciation
Residuals vs a second variable (mileage)

Modelling the data, step by step

This idea usually needs an exponential model. See it worked step by step, with a criterion tip at every step: Exponential y = a e^(kx) + c and y = a·bˣ · 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

Record price, age and mileage from 50+ listings of one model on a used-car website.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

A possible outline

Choose one model and trim level.
Collect listings in one period.
Fit and compare models.
Examine residuals against mileage.
Reflect on listing price vs selling price.

Pitfalls that cost marks

Mixing trims, engines or conditions.
Using asking prices without comment.
Too few old cars.

Showing personal engagement

Choose a car your family owns.
Compare two brands' depreciation rates.
Predict your family car's value and check a valuation site.

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	Linear and exponential regression; Logarithmic transformation
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	Linear and exponential regression; Logarithmic transformation
AI HL	Fits, but add an HL technique	Linear and exponential regression; Logarithmic transformation

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: mixing trims, engines or conditions; and using asking prices without comment. Say how each one affects your answer.

Use of mathematics (E)

SL: linear and exponential regression; logarithmic transformation; interpreting the fitted rate as percentage depreciation; residuals vs a second variable (mileage), each explained in your own words and carried out correctly.

How finance explorations reach the top bands in general: Finance & economics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Fit a model with both age and mileage and compare its predictions.

Extending it for HL

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

See a complete IA, marked

Should I lease or buy my first car? (AI SL) explores a question like this one, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

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

What mathematics does this IA use?

Linear and exponential regression, logarithmic transformation, interpreting the fitted rate as percentage depreciation and residuals vs a second variable (mileage).

Where can I get data for this IA?

Record price, age and mileage from 50+ listings of one model on a used-car website. Checked sources and tools: Desmos graphing calculator.

Turn this idea into your IA

The planner opens with “How fast do cars lose value? Modelling depreciation from listings” filled in: the question, the maths and the data sources.

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