Key Idea: This topic is about the shape of the trig waves and how the numbers in y = a sin(b(x − c)) + d stretch and slide them. Reading a, b, c, d off a given graph is the key skill — it comes up on both papers.
〰️ The three graphs
Amplitude = (max − min) / 2 (half the height of the wave) and the midline / principal axis sits at y = (max + min) / 2. Amplitude applies to sin and cos only — tan has none.
🎛️ The four parameters
- amplitude is |a| — vertical stretch (height of the wave)
- sets the period: 360°/b (or 2π/b) — bigger b = faster wave
- horizontal (phase) shift — moves the wave right by c
- vertical shift — the midline (principal axis) is y = d
✏️ IB-style worked examples
IB-style question — period and amplitude of a wave
A sine-type curve has a maximum of 6 and a minimum of −6 and repeats every 720°. State its amplitude and period.
Step by step:
Amplitude = (max − min) / 2.
Period = the repeat length, read straight off.
Amplitude 6; period 720°.
IB-style question — find a, b, c, d from a graph (Paper 1)
A curve y = a sin(b(x − c)) + d has maximum 9 and minimum 1, period 180°, and its first maximum is at x = 60°. Find a, b, c and d.
Step by step:
a = (max − min)/2, d = (max + min)/2.
b = 360° ÷ period.
A plain sine peaks a quarter-period in (here at 45°); the peak is at 60°, so shift right 15°.
a = 4, b = 2, c = 15°, d = 5.
IB-style question — set up a sinusoidal model
A Ferris wheel's height oscillates between a maximum of 23 m and a minimum of 3 m, completing one turn every 30 s. Find a, d and b (in radians) for h = a sin(bt) + d.
Step by step:
a = (max − min)/2, d = (max + min)/2.
b = 2π ÷ period.
a = 10, d = 13, b = π/15 (one cycle every 30 s).
Important: b sets the period, it isn't the period. The period is 360°/b (or 2π/b) — a larger b makes a shorter, faster wave. And the inside shift flips sign: (x − c) moves the graph right by c, not left.
Tap each card to reveal the answer.
Exam Tips
- sin & cos: range [−1, 1], period 360° (2π). tan: period 180° (π), asymptotes, no amplitude.
- From a graph: amplitude a = (max − min)/2, midline d = (max + min)/2.
- Period P first, then b = 360°/P (or 2π/P) — b is not the period itself.
- (x − c) shifts the wave RIGHT by c; check with max = d + |a|, min = d − |a|.
- Paper 2: set the angle unit (usually radians), graph the model with the target line, and use intersect — check for two hits per cycle.