Modelling with functions

IB Mathematics AI HL· Topic 2: Functions, 2.12 Modelling· 35 min read

1. Selecting the Right Function Type★★☆☆☆⏱ 10 min

The first step in any modelling problem is identifying which family of functions matches the trend and expected behavior of your data. You can identify this by plotting the data and matching its shape to known function behaviors.

📘 Definition

Function Model

A mathematical function constructed to approximate the behavior of a real-world phenomenon, with parameters adjusted to fit observed data.

Linear models: Constant rate of change, straight line trend
Quadratic models: Single turning point, constant acceleration
Exponential models: Constant proportional growth/decay
Logistic models: Growth that levels off at a carrying capacity
Trigonometric models: Periodic repeating behavior

📐 Worked Example

A barista records the temperature of a cooling latte over 20 minutes. The temperature decreases toward room temperature, with the rate of decrease slowing over time. What function type is most appropriate for this model?

1
Identify the key behavior: temperature decreases and approaches a horizontal asymptote at room temperature
2
Compare to common function types: exponential decay approaches a constant asymptote, which matches the observed behavior
3
Eliminate inappropriate options: linear would decrease at a constant rate (and eventually go below room temperature), quadratic has a turning point, and trigonometric is periodic, so none fit

Exam tip:

Always plot your data first before choosing a model; visual inspection eliminates most unsuitable options immediately.

2. Fitting Models to Data★★☆☆☆⏱ 15 min

✓ Calculator OK

For IB AI HL, you will almost always use your graphing display calculator (GDC) or other technology to fit a regression model to a set of data points. The process follows a standard, repeatable structure.

Enter your independent and dependent data into two separate lists on your GDC
Select the pre-determined appropriate function type from the regression menu
Record the fitted parameter values and the coefficient of determination $R^{2}$

📘 Definition

Coefficient of Determination

R^2

A statistic that measures how much variation in the dependent variable is explained by the model. Values range from 0 to 1, with values closer to 1 indicating a better fit.

📐 Worked Example

The following data gives the height of a plant (cm) after $x$ weeks: $(1, 3.1), (2, 5.2), (3, 8.9), (4, 15.1), (5, 26.3)$ . Fit an exponential model of the form $h = a b^{x}$ to this data.

1
Enter $x$ values into list L1 and $h$ values into list L2 on your GDC
2
Select exponential regression from the statistics menu
3
Record the output parameters: $a \approx 1.82$ , $b \approx 1.65$
4
Write the final fitted model:
5
$h = 1.82(1.65)^x$

3. Interpreting Parameters and Predictions★★★☆☆⏱ 15 min

Once you have a fitted model, you need to interpret the meaning of each parameter in context, and use the model to predict values for inputs not in your original data.

📐 Worked Example

For the exponential plant growth model $h = 1.82 (1.65)^{x}$ , interpret the parameter $b$ and predict the height after 6 weeks.

1
Recall that for $y = a b^{x}$ , $b$ is the growth factor per unit of $x$
2
Context interpretation: $x$ is time in weeks, so $b = 1.65$ means the plant's height increases by 65% (a factor of 1.65) each week
3
Substitute $x = 6$ into the model to get the prediction:
4
$h = 1.82(1.65)^6 \approx 35.0$
5
The predicted height after 6 weeks is approximately 35 cm. This is extrapolation (beyond the original data range of 1-5 weeks), so it should be treated as approximate.

4. Assessing Model Validity★★★☆☆⏱ 10 min

No model is perfect; you always need to assess whether your model is appropriate for the context you are studying, even if it has a high $R^{2}$ value.

Check the $R^{2}$ value: values above 0.9 usually indicate a very good fit
Plot the fitted curve over your data points to check for systematic deviations
Check if the model makes long-term sense in context

📐 Worked Example

A researcher fits a linear model to world population data from 1900 to 2000, getting an $R^{2}$ value of 0.98. Why is this model still likely invalid for predicting 2100 population?

1
Recall that linear models predict constant rate of growth forever, with no upper limit
2
Context knowledge: the Earth has finite resources, so population growth will eventually slow and level off
3
The linear model fits the 1900-2000 range well, but extrapolating to 2100 ignores real-world resource constraints

5. Common Pitfalls

Wrong move:

Treating extrapolated predictions as equally reliable as interpolated predictions

Why:

Extrapolation assumes the trend observed in the original data continues outside that range, which is often not true

Correct move:

Always note when a prediction is extrapolation, and comment on its potential unreliability in your answer

Wrong move:

Choosing a function type based only on high $R^{2}$ without checking context

Why:

Higher-order polynomials can always get a higher $R^{2}$ by overfitting to random noise, but will not make sense in context

Correct move:

First select function types based on expected behavior from context, then compare $R^{2}$ between contextually appropriate options

Wrong move:

Forgetting to interpret parameters in context, only stating their numerical values

Why:

Exam questions explicitly ask for context interpretation, and you will lose marks for only giving numbers

Correct move:

Always link the parameter to the variables in the problem, including units where appropriate

Wrong move:

Ignoring impossible outputs from a model (e.g. negative temperature)

Why:

Most models are only valid for a restricted domain of inputs

Correct move:

Always state the domain of validity for your model, and discard impossible outputs as invalid

6. Quick Reference Cheatsheet

Function Type	Common Use Case	Standard Form
Linear	Constant rate of change	$y = m x + c$
Quadratic	Single turning point / constant acceleration	$y = a x^{2} + b x + c$
Exponential	Proportional growth/decay	$y = a b^{x}$
Logistic	Growth leveling off at carrying capacity	$y = \frac{L}{1 + k e ^{- r x}}$
Trigonometric	Periodic repeating behavior	$y = a sin (b x + c) + d$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Fit exponential model to population data
2023 · 2
Quadratic model for projectile motion

Going deeper

official syllabusIB AI HL Topic 2 GuideFull syllabus requirements

What's Next

Modelling with functions is a core transferable skill throughout IB Mathematics AI HL, and it appears in every major topic from statistics to calculus to kinematics. The skills you learn here — selecting appropriate models based on context, fitting parameters to observed data, interpreting results, and assessing validity — will be applied repeatedly when solving real-world problems across the entire syllabus. Modelling also forms the basis for many extended essay and internal assessment projects in IB Math, so mastering these foundations will pay off for your coursework as well as your exams.