Phase Shift Calculator – Guide & Formulas
Calculate the phase shift of sinusoidal functions y = a·sin(bx - c) + d instantly. Find horizontal shift in degrees or radians with step-by-step results.
The Phase Shift Calculator determines the horizontal shift of sinusoidal functions of the form y = a·sin(bx - c) + d or y = a·cos(bx - c) + d. Enter the coefficients and instantly receive the phase shift in both degrees and radians, along with amplitude, period, and a step-by-step explanation.
Key Takeaway
Use the free Phase Shift Calculator to calculate the phase shift of sinusoidal functions y = a·sin(bx - c) + d instantly. find horizontal shift in degrees or radians with step-by-step results. Get instant results with step-by-step explanations.
How to Use the Phase Shift Calculator
- Enter the coefficient a (amplitude multiplier).
- Enter the coefficient b (frequency multiplier).
- Enter the coefficient c (phase constant).
- Enter the constant d (vertical shift).
- Select sine or cosine as the base function.
- View the phase shift, amplitude, period, and vertical shift results.
The Formula
Variable Definitions
- a: Amplitude multiplier (vertical stretch/compression)
- b: Frequency multiplier (horizontal stretch/compression)
- c: Phase constant (determines horizontal shift)
- d: Vertical shift (midline offset)
- c/b: Phase shift: positive = right, negative = left
- 2π/|b|: Period: one complete cycle length
Example: y = 3·sin(2x - π)
Find the phase shift, amplitude, period, and vertical shift for y = 3·sin(2x - π).
- Step 1: Identify coefficients: a = 3, b = 2, c = π, d = 0.
- Step 2: Phase Shift = c/b = π/2 ≈ 1.5708 radians = 90°.
- Step 3: Amplitude = |a| = |3| = 3.
- Step 4: Period = 2π/|b| = 2π/2 = π ≈ 3.1416 radians = 180°.
- Step 5: The graph shifts 90° to the right compared to y = 3·sin(2x).
Frequently Asked Questions
What is phase shift?
Phase shift is the horizontal displacement of a sinusoidal function from its standard position. For y = a·sin(bx - c) + d, the phase shift equals c/b. A positive phase shift moves the graph right, while a negative shift moves it left.
How do I calculate phase shift from an equation?
Rewrite the equation in the form y = a·sin(b(x - h)) + d by factoring out b. The value h = c/b is the phase shift. For example, y = sin(2x - π) can be rewritten as y = sin(2(x - π/2)), giving a phase shift of π/2 radians or 90°.
What is the difference between phase shift and phase angle?
Phase shift is the horizontal displacement measured in the same units as x (often radians or degrees). Phase angle is the phase shift expressed in angular units: φ = c (the numerator before dividing by b). Phase shift = phase angle / b.
Does phase shift apply to cosine functions?
Yes. The phase shift formula c/b works identically for y = a·cos(bx - c) + d. The cosine function simply starts at its peak instead of at zero, but the horizontal shift calculation is the same.
What happens when b is negative?
When b is negative, use the absolute value for the period: Period = 2π/|b|. The phase shift is still c/b, which will flip the sign. For graphing, you can factor out the negative sign to interpret the shift correctly.
How do I find phase shift from a graph?
Identify where the function crosses its midline going upward (for sine) or reaches its peak (for cosine). The horizontal distance from the origin to that point is the phase shift. Compare this to the standard function to determine direction.
What is the relationship between phase shift and period?
Phase shift tells you how far the graph is horizontally displaced, while period tells you the length of one complete cycle. The phase shift as a fraction of the period indicates how much of a cycle the graph has shifted: phase shift / period.
Can phase shift be larger than the period?
Yes. Phase shift can be any real number. However, since the function repeats every period, a phase shift equal to one period produces the same graph. You can always reduce the phase shift modulo the period.
How do phase shifts affect real-world signals?
In electrical engineering and physics, phase shifts represent time delays in alternating current circuits, sound waves, and electromagnetic signals. A phase shift of 180° (π radians) inverts the signal, while 90° shifts it to its quadrature component.
What is the vertical shift d in the formula?
The vertical shift d moves the entire graph up or down. It represents the midline of the sinusoidal function. The function oscillates between d + |a| and d - |a|. When d = 0, the function oscillates around the x-axis.