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
| Graph | Period | Range | Key features |
|---|---|---|---|
| y = sin x | 360° (2π) | −1 ≤ y ≤ 1 | starts at (0, 0) rising; max at 90°, min at 270° |
| y = cos x | 360° (2π) | −1 ≤ y ≤ 1 | starts at (0, 1); max at 0°/360°, min at 180° (= sin shifted left 90°) |
| y = tan x | 180° (π) | all real numbers | no max/min, no amplitude; vertical asymptotes where cos x = 0 (90°, 270°, …) |
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
| Parameter | Effect on the graph | Read it off the graph by… |
|---|---|---|
| a (amplitude) | stretches the wave taller / shorter; amplitude = |a| | (max − min) / 2 |
| b | changes the period to 360°/b (2π/b) — bigger b squeezes it | measure the period P, then b = 360°/P (or 2π/P) |
| c (phase shift) | slides the wave right by c (sign is the opposite of inside) | how far a start-point has shifted sideways |
| d (vertical shift) | lifts the whole wave; the midline is y = d | (max + min) / 2 |
✏️ 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).
🔒 GDC walkthrough
Step through the exact calculator keystrokes, screen by screen, in study mode.
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.
Period of y = tan x? 180° (π) — half the sin/cos period, with asymptotes where cos x = 0.
Amplitude and period of y = 2 cos(4x)? Amplitude 2, period 90° — |a| = 2 and 360°/4 = 90°.
Wave with max 11 and min 3 — find a and d a = 4, d = 7 — a = (11 − 3)/2, midline d = (11 + 3)/2.
A graph has period 720°. What is b? b = ½ — b = 360°/720°.
Transformations in y = sin(x − 40°) + 6? Right 40°, up 6 (amplitude 1, period 360°) — (x − 40°) shifts right; + 6 lifts the midline to y = 6.
Does y = tan x have an amplitude? No — tan is unbounded, so amplitude applies only to sin and cos.
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.