Phase Shift Calculator

Analyze trigonometric functions y = A sin(Bx + C) + D to find phase shift, period, frequency, and more.

Enter Function Parameters

y = 2 sin(3x + 1) + 0

Results

Phase Shift
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--
Amplitude |A| --
Period --
Frequency --
Vertical Shift --
Midline --
Maximum Value --
Minimum Value --

Step-by-Step Solution

Understanding Phase Shift

Phase shift refers to the horizontal displacement of a trigonometric function from its standard position. For a function in the form y = A sin(Bx + C) + D, the phase shift is calculated as -C/B. A positive result means the graph shifts to the right, while a negative result means it shifts to the left.

Components of Trigonometric Functions

Amplitude (A)

The maximum displacement from the midline. |A| determines the height of the wave.

Amplitude = |A|

Period

The length of one complete cycle of the function.

Period = 2pi / |B|

Phase Shift

The horizontal translation of the function.

Phase Shift = -C / B

Vertical Shift (D)

The vertical translation of the function up or down.

Midline: y = D

Frequency

The number of complete cycles per unit interval.

f = |B| / (2pi) = 1 / Period

Range

The set of all possible output values for sine and cosine.

[D - |A|, D + |A|]

How Phase Shift Works

The general form y = A sin(Bx + C) + D can be rewritten as y = A sin(B(x + C/B)) + D. The term C/B represents the horizontal shift. When -C/B is positive, the graph moves to the right; when negative, it moves to the left.

Examples

  • y = sin(x - pi/2): Phase shift = pi/2 to the right (C = -pi/2, B = 1)
  • y = 3cos(2x + pi): Phase shift = -pi/2 to the left (C = pi, B = 2)
  • y = -2sin(4x) + 1: No phase shift (C = 0), but vertical shift up by 1

Applications

Phase shift is fundamental in signal processing, electrical engineering (AC circuits), sound waves, light waves, and any phenomenon modeled by periodic functions. Understanding phase relationships is essential for analyzing interference, resonance, and wave superposition.