IA modelling, step by step · Exponential
Exponential model, step by step: y = a e^(kx) + c and y = a·bˣ
Growth or decay by the same factor in each step — cooling, depreciation, radioactive or drug decay — is exponential. Take logs to turn it into a straight line; if the context has a non-zero asymptote (room temperature), subtract it first.
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
- Equal steps in x multiply y (or y − c) by roughly the same factor.
- The curve approaches a horizontal line (the asymptote) in one direction.
- ln y (or ln(y − c)) against x is close to straight.
The context
- Cooling or warming towards a surrounding temperature (Newton's law of cooling).
- Growth with a fixed percentage rate: money, early-stage populations or views.
- Decay: drug concentration, radioactive counts, the value of a car.
Course fit: Every course: exponential functions and logarithms are in AA and AI at SL; linearising with logs is AI HL; the asymptote c and least-squares refinement suit HL well.
The example data
A mug of tea cools on a desk in a room kept at 21 °C; a probe records the temperature every 5 minutes.
| i | x: Time (min) | y: Temperature of tea (°C) |
|---|---|---|
| 1 | 0 | 85 |
| 2 | 5 | 72.4 |
| 3 | 10 | 61.5 |
| 4 | 15 | 53.9 |
| 5 | 20 | 46.7 |
| 6 | 25 | 42 |
| 7 | 30 | 37.4 |
| 8 | 35 | 34.5 |
| 9 | 40 | 31.4 |
First try: y = a·e^(kx) (asymptote at 0)
Calculator exponential regression assumes the asymptote is y = 0. Fit it first — then ask whether that makes sense for tea in a 21 °C room.
Step 1 · Linearise with logarithms
Assume the horizontal asymptote is y = 0. If y = aekx + c, then ln y = kx + ln|a|: a straight line in (x, ln y) with gradient k and intercept ln|a|.
Step 2 · Tabulate the sums
| i | x | ln y | x² | x(ln y) | (ln y)² |
|---|---|---|---|---|---|
| 1 | 0 | 4.4427 | 0 | 0.0000 | 19.737 |
| 2 | 5 | 4.2822 | 25 | 21.411 | 18.337 |
| 3 | 10 | 4.1190 | 100 | 41.190 | 16.966 |
| 4 | 15 | 3.9871 | 225 | 59.807 | 15.897 |
| 5 | 20 | 3.8437 | 400 | 76.875 | 14.774 |
| 6 | 25 | 3.7377 | 625 | 93.442 | 13.970 |
| 7 | 30 | 3.6217 | 900 | 108.65 | 13.116 |
| 8 | 35 | 3.5410 | 1225 | 123.93 | 12.538 |
| 9 | 40 | 3.4468 | 1600 | 137.87 | 11.880 |
| Σ | 180.000 | 35.0219 | 5100.00 | 663.181 | 137.218 |
n = 9 points. The transformed values are rounded here; keep them unrounded in your calculator or spreadsheet.
In the full worked example
- The rest of the method: steps 3 to 7
- Better: y = a·e^(kx) + c with c = 21 from the context
- 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.
- Can the growth or decay really continue at the same rate? What would stop it?
- Does the asymptote match a known value (for example room temperature for cooling)?
- Is the doubling time or half-life consistent with what you know about the context?
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 an exponential model
Explorations from our idea list whose mathematics includes this model. Each research question is a starting point to make your own.
- Is a hanging chain a parabola? Catenary versus quadratic models: How much better does a catenary y = a cosh(x/a) + c describe a hanging chain than the best-fitting parabola, and does the difference matter for a real design?
- Does my phone battery really drain in a straight line?: Is my phone's battery percentage over a day of normal use better described by a linear, piecewise-linear or exponential model, and can the best model predict when the phone will die?
- How far does a bouncing ball really travel?: Do the rebound heights of a ball form a geometric sequence, and does the geometric-series prediction of total distance travelled match the measured total?
- Exponential or logistic? Modelling my country's population: Which of an exponential, logistic or polynomial model best fits [country]'s population from 1950 to 2000, and which best predicts the actual population from 2000 to today?
- Is Moore's law still true?: Has the number of transistors on microprocessors doubled every two years since 1971, and is there evidence that the doubling time has changed?
- How many times can you fold a sheet of paper?: Does Britney Gallivan's folding-limit formula predict how many times I can fold strips of different paper, and how does the required length grow with the number of folds?
How this earns marks
A exponential 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
Why do my GDC and Desmos give different exponential models?
A GDC fits a straight line to ln y, which minimises errors in ln y. Desmos by default minimises errors in y itself. Both are valid; they weight the points differently. Turn on Desmos's log mode to match the GDC, and say which you used.
Next steps
- Criterion E: use of mathematicsWhat “maths at the level of the course” means for a model, at SL and HL.
- Criterion D: reflectionJudging a model: residuals, comparisons, limitations and implications.
- Self-check your draftTick through the criteria for your own exploration.
- Get feedback on your write-upCriterion-by-criterion feedback on your draft, with evidence from your own text.
- Exemplar: Skydiver (AA HL)AA HL · a complete exploration that fits this kind of model, marked criterion by criterion.
- Exemplar: Coupled cooling model (AI HL)AI HL · a complete exploration that fits this kind of model, marked criterion by criterion.
Related models: Logistic y = L/(1 + C e^(−kx)) · Power y = a·xⁿ (log–log) · 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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