Quick answer

On graphs, phase shift equals C/B in f(x) = A sin(Bx - C) + D. On the circle, it offsets the input angle before sine or cosine is evaluated.

Formula

  • Graph: phase shift = C / B
  • Identity link: cos(x) = sin(x + π/2)

Introduction

Function transformations unit: translations, stretches, and reflections combine. Phase is the horizontal translation piece for periodic functions.

Unit circle interpretation helps when you forget whether a shift moves left or right: track the starting angle on the circle first, then unfold the graph.

If vocabulary is new, start with what a phase shift is and return here for circle and identity connections.

Numeric checks still belong in the Phase Shift Calculator on our homepage after you choose sine or cosine mode.

Trigonometry connections

Angle displacement inside sin(Bx - C) means you evaluate the trig function at a shifted input before scaling and translating vertically.

Graph movements repeat every period, so a phase change repositions every feature: peaks, midline crossings, and troughs all slide together.

Identity relationships let you rewrite shifted sine as cosine, which is useful when a problem gives one form and your tools expect another.

Graphing phase shift functions turns these identities into sketches you can compare with calculator readouts on the same coefficients.

Formulas and identities to keep nearby

  • Phase shift = C / B
  • Period = 2π / B
  • cos(x) = sin(x + π/2)

Memorize the expanded form rule first, then add identities as shortcuts rather than replacements for understanding.

Tangent phase questions require caution: asymptote spacing follows π/|B|, not 2π/|B|.

Degree mode on calculators does not change the symbolic rule; it only changes how you type angles.

Study workflow for trig class

  1. Start with parent graph. Sketch sin(x) or cos(x) quickly.
  2. Apply B. Mark period 2π/B on the axis.
  3. Apply phase. Shift reference point by C/B.
  4. Apply A and D. Stretch vertically and move midline.
  5. Use identities if needed. Convert between sine and cosine forms.
  6. Verify. Confirm with the Phase Shift Calculator.

Identity example

sin(x - π/2) can be viewed as a cosine-shaped curve with a phase link to -cos(x) depending on convention.

For f(x) = 2 sin(x - π/3) + 1, phase shift is π/3, amplitude 2, vertical shift 1, period 2π.

Plotting one cycle after marking x = π/3 prevents drawing the correct shape in the wrong location.