Quick answer

Phase shift is the horizontal translation of a sinusoidal graph. In f(x) = A sin(Bx - C) + D, phase shift = C / B when B is not zero.

Formula

  • Phase shift = C / B
  • Horizontal shift along the x-axis
  • Separate from amplitude A, period 2π/B, and vertical shift D

Introduction

If you can describe amplitude as how tall a wave is and period as how long one cycle takes, phase shift answers a third question: where does the cycle start? That starting offset is what textbooks call horizontal transformation or wave displacement.

The algebra looks small (one fraction, C divided by B) but the graph effect is visible immediately once you plot a single cycle. Many students lose points by reporting C instead of C/B, especially when problems write sin(B(x - h)) first.

When you are ready for numeric readouts, open the Phase Shift Calculator on our homepage and enter A, B, C, and D in the form f(x) = A sin(Bx - C) + D.

After you can define the term in your own words, the phase shift formula guide walks through both standard equation forms side by side, which clears up most sign and factoring confusion.

Definition, meaning, and horizontal transformation

Definition: phase shift is how far you move a sinusoidal graph left or right along the x-axis while keeping its shape. Meaning: it tells you which part of the repeating pattern appears first at a given input value.

Horizontal transformation language fits because only the input x changes in a pure phase adjustment. Amplitude stretches height, period stretches width, vertical shift moves the midline, but phase repositions the wave along the axis.

Wave displacement appears in physics when two sensors record the same frequency but peaks arrive at different times. The math is still C/B once you write the model in standard form.

Students who struggle with vocabulary often improve quickly after working through how to calculate phase shift with one equation rewritten step by step, because the division step forces you to name each parameter correctly.

Standard form reminder

  • f(x) = A sin(Bx - C) + D
  • f(x) = A cos(Bx - C) + D
  • Phase shift = C / B
  • y = A sin(B(x - C)) + D uses C directly before expanding

Both sine and cosine use the same horizontal rule once you match the Bx - C layout. The cosine graph looks different, but C and B still control the slide.

If a problem states sin(B(x - 3)), expand to sin(Bx - 3B) before you enter values in a tool that expects the expanded C.

Graph interpretation: locate x = C/B on the axis, then plot one period of length 2π/B from that anchor.

How to recognize phase shift in a problem

  1. Read the full equation. Identify whether the model is sine or cosine and whether the argument is factored.
  2. Expand if needed. Rewrite B(x - h) as Bx - Bh so C is visible.
  3. List A, B, C, D. Keep angle units consistent across C and the period formula.
  4. Compute C/B. That quotient is the phase shift for the expanded standard form.
  5. Sketch one cycle. Confirm direction with your course convention for left vs right shifts.
  6. Verify numerically. Compare with the Phase Shift Calculator readout row labeled phase shift.

Worked example

Consider f(x) = 3 sin(2x - π) + 1. Here A = 3, B = 2, C = π, and D = 1.

Phase shift = π/2 radians. Period = 2π/2 = π. Amplitude = 3 and vertical shift = 1 describe height and midline separately from the horizontal move.

If the problem had been written sin(2(x - π/2)), expanding first gives the same B = 2 and C = π, so the shift still equals π/2.