a stretches; b squeezes the period: In y = a sin(bx): a is the amplitude (vertical stretch), and b changes the period to 360°/b (or 2π/b). A bigger b means a shorter, faster wave.
IB-style question — read a and b
State the amplitude and period of y = 3 sin(2x) (x in degrees).
Step by step
- Amplitude is |a|.
- Period is 360°/b.
Final answer
Amplitude 3, period 180°.
b divides the period: Period is 360°/b — a bigger b gives a smaller period (the wave repeats faster).
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d raises the wave; find a and d from max & min: y = a sin(bx) + d lifts the whole wave by d (the midline is y = d). From a model's max and min: a = (max − min)/2 and d = (max + min)/2.
IB-style question — a and d from a model
A tide height oscillates between a maximum of 7 m and a minimum of 1 m. Find a and d for the model h = a sin(bt) + d.
Step by step
- Amplitude a.
- Vertical shift d (midline).
Final answer
a = 3, d = 4 (so the wave swings 3 either side of the midline 4).
max = d + a, min = d − a: Check: the maximum is d + a and the minimum is d − a — a quick way to verify your values.
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c slides the wave sideways: In y = a sin(b(x − c)) + d, the c is a horizontal (phase) shift — the wave moves right by c (the inside change is the opposite of its sign, as with all transformations).
IB-style question — identify all four
Describe the transformations in y = 2 sin(x − 30°) + 5.
Step by step
- a = 2 (amplitude), inside (x − 30°) → right 30°, + 5 → up 5.
Final answer
Amplitude 2, shifted right 30° and up 5 (period unchanged, b = 1).
Order of reading: Read amplitude (a), period (from b), then the shifts: right c, up d. Match each to the right part of the equation.
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Real oscillations: tides, springs, Ferris wheels: Periodic real-world quantities are modelled by y = a sin(b(x − c)) + d: read a and d from the max & min, and b from the period (b = 360°/period or 2π/period).
IB-style question — find b from the period
A Ferris wheel's height repeats every 40 seconds. Find b (in radians) for h = a sin(bt) + d.
Step by step
- Period = 2π/b, so b = 2π/period.
Final answer
b = π/20 (so the wave completes one cycle in 40 s).
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Solve on the GDC: On Paper 2, once the model is set up, graph it and use intersect to find when the quantity hits a target value (links to 3.8).