Phase Shift Calculator
Calculate amplitude, period, phase shift, and vertical shift for trigonometric functions
Function Parameters
Function Equation
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
Function Evaluation
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
Parameter Effects
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