Other Models (AI SL)
Not every real-world pattern is a straight line, a parabola or an exponential curve. This topic covers the remaining model types in the AI SL toolkit: sinusoidal models \(f(x)=a\sin(bx)+d\) for periodic phenomena like tides and Ferris wheels, cubic models for situations with two turning points, and piecewise models built from different formulas over different intervals. Together they round out the modelling process - choosing, fitting and reading off features from whichever function type fits the data.
What the syllabus says
This topic maps onto the modelling section of the official IB Applications & Interpretation syllabus.
| Code | Syllabus content |
|---|---|
| SL2.5 | Two of several modelling types in this section: cubic functions \(f(x)=ax^3+bx^2+cx+d\), and sinusoidal functions \(f(x)=a\sin(bx)+d\), \(f(x)=a\cos(bx)+d\). Students find amplitude \(a\), period \(\tfrac{360}{b}\), or the equation of the principal axis \(y=d\), but are not expected to translate between \(\sin x\) and \(\cos x\). Suggested contexts: tides, weather patterns, motion of Ferris and bicycle wheels, annual temperatures. (This section also covers linear, quadratic, exponential and direct/inverse variation models.) |
| SL2.6 | Modelling skills: use the modelling process (fit a function to data or a description, use it to make predictions, and comment on its validity and limitations) with the theoretical models introduced in SL2.5, including piecewise-defined models built from these function types. |
Sinusoidal, cubic and piecewise modelling questions appear on both Paper 1 and Paper 2, often alongside GDC graphing.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is a sinusoidal model?
A sinusoidal model is a function of the form \(f(x)=a\sin(bx)+d\) or \(f(x)=a\cos(bx)+d\), used for quantities that rise and fall in a repeating cycle, such as tides or the height of a point on a wheel.
e.g. The depth of water at a harbour is modelled by \(D(t)=6+4\sin(30t^\circ)\), \(t\) hours after midnight.
What is amplitude?
Amplitude is the coefficient \(a\) in front of the sine or cosine term - half the vertical distance between the maximum and minimum values. It measures how far the graph swings above and below its centre.
e.g. For \(D(t)=6+4\sin(30t^\circ)\), the amplitude is \(4\), so the depth ranges from \(6-4=2\) m to \(6+4=10\) m.
What is the principal axis?
The principal axis (or midline) is the horizontal line \(y=d\) that the graph oscillates evenly about - halfway between the maximum and minimum values.
e.g. \(T(t)=18+6\sin(15t^\circ)\) has principal axis \(T=18\).
What is the period?
The period is the length of \(x\) needed for one complete cycle of the graph. For \(f(x)=a\sin(bx)+d\) with \(x\) in degrees, the period is \(\dfrac{360}{b}\).
e.g. \(T(t)=18+6\sin(15t^\circ)\) has period \(\dfrac{360}{15}=24\).
What is a piecewise model?
A piecewise model uses a different formula on different intervals of the input, joined together at boundary values, for a quantity that behaves in genuinely different ways over different ranges.
e.g. A parking charge is \(C(t)=4\) for \(0
Key formulas
Sinusoidal models are described by their amplitude, principal axis and period. The tables below summarise the standard forms and how each feature is read off.
Formula reference
The sinusoidal and cubic forms, along with the period formula, are listed in the AI formula booklet under functions.
| Formula | Used for | Booklet? |
|---|---|---|
| \(f(x)=a\sin(bx)+d\) | Sinusoidal model (sine form) | ✓ Yes |
| \(f(x)=a\cos(bx)+d\) | Sinusoidal model (cosine form) | ✓ Yes |
| Period \(=\dfrac{360}{b}\) | Length of one full cycle (degrees) | ✓ Yes |
| \(f(x)=ax^3+bx^2+cx+d\) | Cubic model | ✓ Yes |
| Piecewise formula (case-by-case) | Different rule on different intervals | Not in booklet - defined per question |
Continuous models vs piecewise models
Sinusoidal and cubic models use one formula everywhere; piecewise models are built from several.
| Feature | Continuous model | Piecewise model |
|---|---|---|
| Defined by | One formula for every \(x\) | A different formula on each interval |
| Example | \(D(t)=6+4\sin(30t^\circ)\) | \(C(t)=4\) for \(0 |
| Graph shape | One smooth, repeating curve | Joined segments, possibly with corners |
| Evaluating | Substitute directly | Identify the correct interval first |
Key features of a sinusoidal model
Every sinusoidal modelling question asks about one or more of these features.
Amplitude
\[a\]
Half the distance between the maximum and minimum values of the model.
✓ In the formula bookletPrincipal axis
\[y=d\]
The horizontal line the graph oscillates about - halfway between max and min.
✓ In the formula bookletPeriod
\[\frac{360}{b}\]
How much the input needs to change for the graph to complete one full cycle.
✓ In the formula bookletMaximum & minimum
\[d+a\ \text{(max)}\qquad d-a\ \text{(min)}\]
The extreme values of the model, found directly from the amplitude and principal axis.
Not in the formula booklet - derived directlyWorking with a piecewise model
Piecewise questions are less about formulas and more about correctly choosing which formula to use.
Choosing the right piece
Always compare the given input to the interval boundaries stated in the question before deciding which formula to substitute into.
Evaluating at a boundary
Check carefully whether the boundary point itself belongs to the piece above or below it - the inequality signs (\(<\) vs \(\le\)) tell you exactly which.
Solving within a piece
To find when the model reaches a target value, set the relevant piece equal to that value and check the solution actually falls inside that piece's interval.
Worked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
A model is \(T(t)=18+6\sin(15t^\circ).\)
(a) State the amplitude.
(b) State the principal axis (midline).
(c) State the period.
Worked solution
(a) Amplitude. The coefficient of the sine is the amplitude: \(6\). A1
(b) Principal axis (midline). The constant added is the midline: \(T=18\). A1
(c) Period. For \(\sin(bt^\circ)\) the period is \(\dfrac{360}{b}=\dfrac{360}{15}=24\). A1
The depth of water at a harbour is \(D(t)=6+4\sin(30t^\circ),\) \(t\) hours after midnight.
(a)(i) State the maximum depth.
(a)(ii) State the minimum depth.
(b) Find the depth at \(t=4.\)
(c) State the period of the model.
Worked solution
(a)(i) The amplitude is \(4\) about the midline \(6\): max \(=6+4=10\) m. A1
(a)(ii) Min \(=6-4=2\) m. A1
(b) Depth at \(t=4\). Angle \(=30\times4=120^\circ\), \(\sin120^\circ\approx0.86603\): M1
\(D(4)=6+4(0.86603)\approx9.46\text{ m}.\) A1
(c) Period. \(\dfrac{360}{30}=12\) hours. A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Leaving the GDC in radian mode. AI sinusoidal models are written with the angle in degrees, like \(\sin(30t^\circ)\) - a calculator left in radian mode gives a completely different, wrong value with no warning.
- Confusing amplitude with the maximum value. The amplitude is half the range - the maximum is amplitude plus the principal axis, not the amplitude on its own.
- Substituting into the wrong piece of a piecewise model. Always check the input against the stated interval boundaries first - using the wrong formula gives a plausible-looking but wrong answer.
- Assuming a period formula applies to every trig model. The period formula \(360/b\) only works when the angle is in degrees and the coefficient of \(x\) is exactly \(b\) - check the model is in the standard form first.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
The heart of AI modelling - find the best-fitting curve for a data set, not just a straight line.
- Enter the data in two lists (x and y).
- STAT → EDIT to enter L1/L2, then STAT → CALC → QuadReg / CubicReg / ExpReg / PwrReg / SinReg.TI-84
- In a Lists & Spreadsheet page enter the data, then menu → Statistics → Stat Calculations → choose the regression type.Nspire
- Main menu → Statistics, enter the data in lists, then CALC → REG and pick X² / X³ / Exp / Power / Sin.Casio
Tip: Turn DiagnosticOn (TI-84: 2nd → 0 → DiagnosticOn) to see R². Choose the model with the best R² that also makes sense for the context - remember to set degree mode before graphing a sinusoidal fit.
After fitting several models (linear, quadratic, exponential...), you need to decide which fits the data best - R² is the key tool.
- Fit each candidate model in turn and note the R² value each time.
- Turn DiagnosticOn first (2nd → 0, scroll to DiagnosticOn, ENTER) - then R² appears after every regression.TI-84
- R² is shown automatically after each regression calculation in the Statistics menu.Nspire
- R² (displayed as r²) appears in the regression output; run CALC → REG for each model type and compare.Casio
- The model with R² closest to 1 explains the most variation in y - but also consider whether the model makes sense for the context.
Tip: R² alone doesn't tell you whether the model is appropriate - always look at the scatter plot too. A high R² on a sinusoidal model fitted to steadily growing data is meaningless.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Sinusoidal, cubic and piecewise modelling questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
What's the difference between amplitude and the principal axis?
The amplitude \(a\) is half the vertical distance between the maximum and minimum - how far the graph swings above and below its centre line. The principal axis \(y = d\) is that centre line itself. Together, max \(= d + a\) and min \(= d - a\).
How do I find the period of a sinusoidal model?
For \(f(x) = a \sin(bx) + d\) with \(x\) in degrees, the period is \(360/b\). With \(x\) in radians it's \(2\pi/b\). The period is how much \(x\) has to increase for the graph to complete one full cycle and repeat.
Why do I need degree mode on my GDC for these models?
IB AI sinusoidal models are usually written with the angle in degrees, like \(\sin(30t^\circ)\). If your calculator is set to radian mode it will evaluate the same expression completely differently, giving a wrong answer without any error message - always check the mode before you graph or evaluate.
How do I evaluate a piecewise model at a given input?
First check which interval the input falls into, then use only the formula defined for that interval. A common mistake is plugging the value into the wrong piece - always compare the input to the boundary values stated in the question before substituting.
Sub-topics
Other Models broken down into its individual skills, each with its own focused page.
Related topics
More Functions & Modelling topics from the same AI SL syllabus unit, in case you want to keep going.