The big idea: Use a sinusoidal model y = a·sin(bx) + d when the situation repeats in a regular cycle — tides, temperature, hours of daylight, rotating wheels.
| Parameter | What it controls | Formula to find it |
|---|---|---|
| a (amplitude) | Half the total height (max − min)/2 | a = (max − min) / 2 |
| d (vertical shift) | Midline — the average of max and min | d = (max + min) / 2 |
| b | Controls the period | b = 2π / period |
Larger amplitude (a)
- Taller wave — bigger peaks and deeper troughs
- a = (max − min) / 2
- Example: a = 3 means graph goes 3 units above and below midline
Larger period (smaller b)
- Wider wave — takes longer to complete one cycle
- Period = 2π / b
- Example: Period = 12 hours → b = π/6
Common contexts: Tides, temperature variation through the year, daylight hours — these all repeat in a cycle. If a problem mentions 'every 12 hours' or 'annual cycle', think sinusoidal model.
The big idea: Given the maximum and minimum values and the period, you can find all four parameters of a sinusoidal model. Work step by step: find a → find d → find b → write the model.
Building a sinusoidal model from context
Tidal height (m) repeats every 12 hours. Maximum height = 8 m, minimum height = 2 m. Write a model for height H in terms of time t (hours), assuming H starts at the midline and increases.
Step by step
- Find the amplitude a.
- Find the vertical shift d.
- Find b from the period.
- Write the model.
Final answer
H = 3 sin(πt/6) + 5
Work step by step: IB awards marks for each correctly identified parameter. Show each step — find a, then d, then b separately. Missing one step costs marks even if the final equation is correct.
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The big idea: Students most often confuse amplitude with the maximum value, and mix up the formula for b. The amplitude is half the total range — not the maximum value. And b = 2π/period — not 1/period.
Wrong
- Amplitude = maximum value (e.g. a = 8)
- b = 1/period (forgot the 2π)
- d = 0 when there is a vertical shift
- Period = b (reversed the formula)
Correct
- a = (max − min)/2 = (8 − 2)/2 = 3
- b = 2π/period
- d = (max + min)/2 = midline height
- period = 2π/b → rearrange to get b
Check with the formula: After finding a, d, and b, verify: does your model give the correct max (d + a) and min (d − a)? This quick check catches sign errors.
The big idea: Once you have the sinusoidal model, substitute values of t to find H (or whatever the output is). Use your GDC to solve sinusoidal equations — set y = target value and find the intersections.
Prediction using a sinusoidal model
Using H = 3sin(πt/6) + 5, find the height at t = 3 and find all t in [0, 24] when H = 6.
Step by step
- Substitute t = 3.
- For H = 6: set up equation and use GDC.
- Graph both sides and find intersections in [0, 24].
Final answer
H(3) = 8 m. H = 6 at approximately t = 0.6, 5.4, 12.6, 17.4 hours.
Find all solutions in the domain: Sinusoidal equations have multiple solutions per cycle. IB questions often ask for all values in a given interval. Use GDC intersections — do not stop at the first answer.