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Learn Index · Trigonometry Grade 11–12

Amplitude, period, phase shift, and vertical shift

Amplitude is the height, period the length of a cycle, phase shift the horizontal shift, vertical shift the move up or down.

Read

The general sinusoidal function

y = a sin(b(x − c)) + d (or cos)

Parameter Effect
a amplitude = |a|; a < 0 reflects in the midline
b period = 360°/|b| (or 2π/|b|)
c phase shift: right if c > 0
d vertical shift: midline y = d

Reading the parameters

Example: y = 3 sin(2(x − 45°)) + 1

  • Amplitude: 3
  • Period: 360°/2 = 180°
  • Phase shift: 45° right
  • Midline: y = 1
  • Max: 1 + 3 = 4; min: 1 − 3 = −2

Watch the factoring: y = sin(2x − 90°) = sin(2(x − 45°)). The phase shift is 45°, not 90°.

Finding parameters from max and min

  • Amplitude = (max − min)/2
  • Midline: d = (max + min)/2

Example: max 14, min 2 → amplitude 6, midline y = 8.

Graphing step by step

y = 2 cos(x − 30°) − 1

  1. Midline y = −1; max 1; min −3.
  2. Period 360°.
  3. Start a cosine cycle at a max, shifted to x = 30°.
  4. Key points: (30°, 1), (120°, −1), (210°, −3), (300°, −1), (390°, 1).
  5. Draw the smooth wave.

Writing an equation from a graph

  1. Find max and min → a and d.
  2. Measure the period → b = 360°/period.
  3. Choose a starting point:
    • Sine: start at the midline going up.
    • Cosine: start at a maximum.
  4. That x-value is the phase shift c.

Example: A graph has max 5, min −1, period 120°, and a maximum at x = 20°.

  • a = 3, d = 2, b = 3, c = 20°
  • y = 3 cos(3(x − 20°)) + 2

Many equations for one graph

The same graph can be written with sine or cosine, and with different phase shifts (differing by whole periods). All are correct if they match the graph.

Applications

Every transformation has a real meaning in models:

  • Amplitude: the size of a tide’s swing
  • Period: the time for one cycle (12.4 h for many tides)
  • Phase shift: when the cycle starts
  • Midline: the average level

Worked examples

Example 1: Period of y = sin(4x)? 360°/4 = 90°.

Example 2: Amplitude and midline of y = −5 cos x + 3? 5; y = 3.

Example 3: Phase shift of y = sin(3x + 60°)? 3(x + 20°) → 20° left.

Common mistakes

  • Using b as the period instead of 360°/b.
  • Not factoring b out before reading the phase shift.
  • Giving a negative amplitude.

Try it yourself

  1. y = 4 sin(x/2) − 2: amplitude, period, midline?
  2. Max 9, min 1: amplitude and midline?
  3. Phase shift of y = cos(2x − 60°)?
  4. Write an equation: amplitude 2, period 180°, midline y = 0, starting at a max at x = 0.

Answers: 1) 4, 720°, y = −2 2) 4, y = 5 3) 30° right 4) y = 2 cos(2x)

Quick recap

  • y = a sin(b(x − c)) + d.
  • Amplitude |a|, period 360°/|b|, phase shift c, vertical shift d.
  • Factor b out before reading c.