Phase Shift Calculator

Calculate amplitude, period, phase shift, and vertical shift for trigonometric functions

Function Parameters

Function Equation

f(x) = sin(x)
Standard form: f(x) = A × sin(Bx - C) + D

Controls the height of the wave

Controls the period of the wave

Affects horizontal shift (C/B)

Moves the wave up or down

Results

Function Properties

1.000
Amplitude
6.283
Period (radians)
0.000
Phase Shift (no shift)
0.000
Vertical Shift

Function Evaluation

f(0) = 0.000000
Function value at x = 0
Frequency Information

Angular frequency: 1 rad/unit

Frequency: 0.15915494309189535 Hz

Range Information

Range: [-1.00, 1.00]

Midline: y = 0

Function Graph

f(x) = sin(x)

Amplitude: 1.00 | Period: 6.28 | Phase Shift: 0.00 (none) | Vertical Shift: 0

Interactive graph visualization would be displayed here in a full implementation

Graph shows: sin function with amplitude 1.00, period 6.28, phase shift 0.00 (none), and vertical shift 0

Example: Phase Shift Analysis

Problem

Find the amplitude, period, phase shift, and vertical shift for f(x) = 0.5 × sin(2x - 3) + 4

Solution

Given: f(x) = 0.5 × sin(2x - 3) + 4

Standard form: f(x) = A × sin(Bx - C) + D

Where: A = 0.5, B = 2, C = 3, D = 4

Amplitude: |A| = |0.5| = 0.5

Period: 2π/|B| = 2π/2 = π

Phase shift: C/B = 3/2 = 1.5 (right)

Vertical shift: D = 4 (upward)

Phase Shift Formulas

Standard Form

f(x) = A × sin(Bx - C) + D

f(x) = A × cos(Bx - C) + D

Calculations

Amplitude = |A|

Period = 2π / |B|

Phase Shift = C / B

Vertical Shift = D

Understanding Phase Shift

Positive: Graph shifts to the right
Negative: Graph shifts to the left
Zero: No horizontal shift
Formula: Phase shift = C/B

Parameter Effects

A (Amplitude):Height of oscillation
B (Frequency):Speed of oscillation
C (Phase Constant):Horizontal shift amount
D (Vertical Shift):Centerline position
Note: Phase shift is the horizontal translation that moves the standard trig function to match your function.

Understanding Phase Shift in Trigonometric Functions

What is Phase Shift?

Phase shift is the horizontal translation of a trigonometric function. It describes how far left or right the standard sine or cosine curve has been moved. The phase shift is calculated as C/B, where C is the phase constant and B is the frequency.

Applications

  • Signal processing and wave analysis
  • Physics: oscillations and harmonic motion
  • Engineering: AC circuit analysis
  • Music and acoustics

Key Relationships

Amplitude: Controls vertical stretch/compression

Period: Inversely related to frequency (B)

Phase Shift: Horizontal translation (C/B)

Vertical Shift: Moves entire graph up/down

Common Transformations

sin(x - π/2) = -cos(x): π/2 right shift

cos(x + π/2) = -sin(x): π/2 left shift

sin(2x - π): π/2 right shift, double frequency