IA modelling, step by step · Reciprocal
Reciprocal model, step by step: y = a/x + b and y = a/(x − h) + k
Inverse relationships — time against speed, cost per person against group size — curve towards two asymptotes. Take the vertical asymptote from the context, plot y against 1/(x − h), and the gradient and intercept of that line give a and k.
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
- y falls steeply at first, then levels off towards a horizontal line (or rises towards one).
- x × y (or (x − h)(y − k)) is roughly constant.
- A plot of y against 1/x is close to straight.
The context
- Time for a fixed distance or task at different speeds or rates.
- Cost per person when a fixed cost is shared.
- Physics: Boyle's law, lens formulas, intensity-type relationships.
Course fit: AA SL and HL (the reciprocal function and rational functions); AI SL (inverse variation, axⁿ with n = −1); AI HL (linearising). Everyone can justify it well because the asymptotes usually have a meaning.
The example data
A cyclist logs the same commute on different days: average speed while moving (from a bike computer) and total door-to-door time, which includes waiting at traffic lights.
| i | x: Average moving speed (km/h) | y: Journey time (min) |
|---|---|---|
| 1 | 12 | 66.5 |
| 2 | 15 | 53.6 |
| 3 | 18 | 46.4 |
| 4 | 20 | 41.8 |
| 5 | 22 | 39.1 |
| 6 | 25 | 34.5 |
| 7 | 28 | 32 |
| 8 | 30 | 30.3 |
Example 1: y = a/x + b (journey time and speed)
Time = distance ÷ speed suggests a/x, and time spent stopped at lights adds a constant b. The vertical asymptote is x = 0 (speed zero), so plot y against 1/x.
Step 1 · Asymptotes from the context, then linearise
The vertical asymptote is taken at x = 0. If y = a/(x − h) + k, then plotting y against X = 1/x gives a straight line with gradient a and intercept k (the horizontal asymptote).
Step 2 · Tabulate the sums
| i | 1/x | y | 1/x² | 1/xy | y² |
|---|---|---|---|---|---|
| 1 | 0.083333 | 66.5 | 0.0069444 | 5.5417 | 4422.25 |
| 2 | 0.066667 | 53.6 | 0.0044444 | 3.5733 | 2872.96 |
| 3 | 0.055556 | 46.4 | 0.0030864 | 2.5778 | 2152.96 |
| 4 | 0.05 | 41.8 | 0.0025 | 2.09 | 1747.24 |
| 5 | 0.045455 | 39.1 | 0.0020661 | 1.7773 | 1528.81 |
| 6 | 0.04 | 34.5 | 0.0016 | 1.38 | 1190.25 |
| 7 | 0.035714 | 32 | 0.0012755 | 1.1429 | 1024 |
| 8 | 0.033333 | 30.3 | 0.0011111 | 1.0100 | 918.09 |
| Σ | 0.410058 | 344.200 | 0.0230280 | 19.0929 | 15856.6 |
n = 8 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 and the residual plot
- Example 2: y = a/(x − h) + k (a lens, asymptotes 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.
- Do the asymptotes have a meaning in your context?
- What does the model predict for very small x, and is that possible?
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 reciprocal model
Explorations from our idea list whose mathematics includes this model. Each research question is a starting point to make your own.
- Does light really follow an inverse-square law?: Does the illuminance measured by my phone fall as 1/d² as I move away from a lamp, and at what distances does the model fail?
- How long does a skydiver take to reach terminal velocity?: Using a differential equation with quadratic air resistance, how long does a skydiver take to reach 95% of terminal velocity, and how different is the answer from a linear-drag model and from Euler's method?
How this earns marks
A reciprocal 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 choose h for y = a/(x − h) + k?
From the context whenever possible — the x value where the quantity becomes undefined or infinite (a focal length, a minimum group size, a threshold). If the context gives no value, try several values of h and choose the one with the smallest SSR, and say so.
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.
Related models: Power y = a·xⁿ (log–log) · Exponential y = a e^(kx) + c and y = a·bˣ · 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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