Quick answer

Rewrite each problem to A sin(Bx - C) + D or cosine form, then phase shift = C/B unless the factored form asks for C directly.

Formula

  • Sine and cosine: phase shift = C / B
  • Check period separately with 2π / B

Introduction

Examples teach pattern recognition faster than memorizing a single template. Notice how the same four parameters appear with different numbers and signs.

Tangent problems sometimes appear in the same unit. Tangent has a different period rule, but the language of horizontal displacement still appears.

Before copying these setups, confirm your form with the phase shift formula so you know whether the final answer should be C or C/B.

Verify every row using the Phase Shift Calculator on our homepage after you finish the algebra by hand.

Types of examples in this guide

Sine function examples highlight positive and negative amplitude with midline shifts.

Cosine function examples often start near a maximum, which helps you see why phase and shape choices matter on graphs.

Waveform and signal examples describe two channels at the same frequency with different peak timing.

When examples feel easy, move to phase shift in trigonometry for identity-based rewrites that connect graphs to the unit circle.

Shared reference lines

  • Phase shift = C / B
  • Period = 2π / B
  • Vertical shift = D

Keep a three-column table: parameter, value, effect on graph.

Separate horizontal and vertical information before you plot.

Real-world signal examples may use time t instead of x; the algebra is identical after renaming the variable.

Pattern for each example

  1. Name the function. Sine, cosine, or a word problem modeled by one of them.
  2. Rewrite. Expand factored arguments.
  3. Compute C/B. Report horizontal shift with units.
  4. List other parameters. Amplitude, period, and D belong in the same solution block.
  5. Interpret. One sentence should explain what the shift means in context.
  6. Verify. Use the Phase Shift Calculator to confirm arithmetic.

Multi-part example set

Sine: f(x) = 2 sin(3x - π/2) + 1 gives phase shift π/6, period 2π/3, amplitude 2, vertical shift 1.

Cosine: g(x) = -cos(2x - π) + 4 gives phase shift π/2, period π, amplitude -1, vertical shift 4.

Waveform: y1 = sin(x) and y2 = sin(x - π/4) differ by π/4 radians in phase at the same frequency.