IB Maths AI SL · Unit 2: Functions
IB Maths AI SL Sinusoidal Modelling Questions
Exam-style IB Maths AI SL sinusoidal modelling questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 10 questions
- Paper 1: 3
- Paper 2: 7
- 2 easy
- 4 medium
- 2 hard
- 2 very hard
- 3 worked examples
Practise Sinusoidal Modelling questions →
AI SL formula booklet
What you need to know
Modelling tides, daylight hours, or Ferris-wheel height with y = a·sin(bx + c) + d. Amplitude, period, and phase shift each carry an A-mark. Sinusoidal models for periodic data overview →
Slope, intercept, and interpreting the gradient in context. SL AI dresses linear models as taxi fares, phone plans, and rental costs — always ask what the gradient MEANS in the real-world context. Linear functions and modelling overview →
What's examined in AI SL sinusoidal modelling
The question bank covers these sinusoidal modelling question types (number of questions in brackets):
- Analysing sinusoidal models (7)
- Constructing sinusoidal models (2)
- Determining model parameters (1)
Sinusoidal Modelling worked examples
Worked example 1: Reading amplitude and principal axis · easy
Water depth in a harbour: $D(t) = 4\cos(30t^\circ) + 12$ ($t$ in hours after midnight). Find the exact amplitude and principal axis.
1. Recall $y = a\cos(bx) + d$: $a$ is amplitude, $d$ is principal axis.
2. Amplitude: coefficient of $\cos$ is $4$.
3. Principal axis: constant term is $12$.
4. State: amplitude $= \mathbf{4\text{ m}}$; principal axis $= \mathbf{y = 12}$.
Examiner tip: Amplitude is a POSITIVE value: the distance from principal axis to max (or min). It is NOT the peak-to-trough distance (which is $2a$).
Worked example 2: Max and min of a sinusoidal temperature model · medium
Greenhouse temperature: $T(t) = 15 - 5\sin(15t^\circ)$ ($t$ in hours). Find the exact maximum and minimum temperatures.
1. Principal axis: constant term $= 15$.
2. Amplitude: $|-5| = 5$.
3. Maximum: principal axis $+$ amplitude $= 15 + 5 = 20$.
4. Minimum: principal axis $-$ amplitude $= 15 - 5 = 10$.
5. State: max $= \mathbf{20^\circ\text{C}}$; min $= \mathbf{10^\circ\text{C}}$.
Examiner tip: Regardless of the sign of the sine/cosine coefficient, max = principal axis + amplitude and min = principal axis − amplitude.
Worked example 3: Time above a threshold — sinusoidal inequality · hard
Using $D(t) = 4\cos(30t^\circ) + 12$, use your GDC to find the total time between midnight ($t = 0$) and noon ($t = 12$) when depth is strictly greater than $14\text{ m}$.
1. Set up: $4\cos(30t^\circ) + 12 > 14$.
2. Set the GDC to DEGREES mode.
3. Graph $Y_1 = 4\cos(30x) + 12$ and $Y_2 = 14$ on $0 \le x \le 12$.
4. Find intersections: $x = 2$ and $x = 10$.
5. Analyse: at $t = 0$ depth $= 16 > 14$; drops below at $t = 2$; rises above again at $t = 10$.
6. Total: $[0, 2]$ ($2\text{ h}$) $+ [10, 12]$ ($2\text{ h}$) $= \mathbf{4\text{ hours}}$.
Examiner tip: Sketch the sine/cos curve from your GDC screen onto paper when solving inequalities — visualising the peaks and troughs prevents you picking the wrong interval.
Try these IB Maths AI SL sinusoidal modelling questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 3 marks · Paper 1
The height of a point on a rotating wheel, \(h\) in metres above the ground, is modelled by the function \(h(t) = 5\sin(30t)^{\circ} + 12\), where \(t\) is the time in seconds.
- Write down the amplitude of the function.
- Write down the equation of the principal axis.
- State the maximum height reached by the point.
Question 2 · medium · 4 marks · Paper 2
The temperature, \(T\) in \(^{\circ}\text{C}\), of a greenhouse is modelled by \(T(t) = 4\cos(15t)^{\circ} + 22\), where \(t\) is the time in hours after midnight.
- Calculate the period of the model and explain what it represents in context.
- Calculate the temperature of the greenhouse at 06:00 (when \(t = 6\)).
Question 3 · hard · 6 marks · Paper 2
The depth of water, \(D\) in metres, in a harbour is modelled by the function \(D(t) = a\cos(bt)^{\circ} + d\), where \(t\) is the time in hours after high tide. High tide occurs at \(t = 0\) with a maximum depth of \(14 \text{ m}\). The next low tide occurs at \(t = 6\) hours with a minimum depth of \(6 \text{ m}\).
- Find the exact values of \(a\) and \(d\).
- Find the value of \(b\).
- Using your Graphic Display Calculator, find the amount of time during the first 12 hours when the depth of the water is strictly greater than \(12 \text{ m}\).
All 10 sinusoidal modelling questions with mark schemes →
FAQ
How many IB Maths AI SL sinusoidal modelling questions are there?
There are 10 exam-style sinusoidal modelling questions in the AI SL question bank (Paper 1: 3 · Paper 2: 7), graded 2 easy, 4 medium, 2 hard, 2 very hard. Every question has a full IB-style mark scheme (M, A and R marks).
Is sinusoidal modelling on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 3 · Paper 2: 7. Practise with your GDC — AI papers expect calculator methods throughout.
Where can I get the mark schemes?
Open the AI SL Unit 2 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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