# Phase Shift Calculator

URL: https://speedycalculator.com/calculators/math/phase-shift/
Last updated: 2026-10-30

> The phase shift calculator analyses y = A·sin(Bx − C) + D or y = A·cos(Bx − C) + D: amplitude, period, frequency, phase shift and vertical shift, and the value of the function at any x.

## Inputs

- Function
- A (amplitude factor)
- B (frequency factor)
- C (phase)
- D (vertical shift)
- Evaluate at x

## Results

- Phase shift C/B
- Amplitude
- Period
- Frequency (cycles per unit)
- Vertical shift
- Maximum value
- Minimum value
- y at x

## Formula

```
Amplitude = |A|,  Period = 2π/|B|,  Phase shift = C/B,  Vertical shift = D
```

## Explanation

Every sinusoidal wave – sound, alternating current, tides, daylight hours – is a stretched and shifted sine. The phase shift says how far the wave is moved horizontally compared with the basic sin x or cos x.

### How to use the Phase Shift Calculator

- **Function**
- **A (amplitude factor)**
- **B (frequency factor)**
- **C (phase)**
- **D (vertical shift)**
- **Evaluate at x**

Results update as you type, and everything is calculated in your browser.

### How it works

Pick sine or cosine and enter A, B, C and D. A positive phase shift moves the graph right, a negative one left. The function value at x is computed directly.

### Inputs explained

FieldWhat to enterFunctionChoose one: y = A·sin(Bx − C) + D, y = A·cos(Bx − C) + DA (amplitude factor)A numberB (frequency factor)A numberC (phase)A numberD (vertical shift)A numberEvaluate at xA number

### What each result means

- **Phase shift C/B** = [C (phase)]÷[B (frequency factor)]
- **Amplitude** = abs([A (amplitude factor)])
- **Period** = 2×pi÷abs([B (frequency factor)])
- **Frequency (cycles per unit)** = abs([B (frequency factor)])÷(2×pi)
- **Maximum value** = [D (vertical shift)] + abs([A (amplitude factor)])
- **Minimum value** = [D (vertical shift)] - abs([A (amplitude factor)])

### Example values

How the main result changes when **A** changes and everything else stays at the example values above:

A (amplitude factor)Phase shift C/B10.51.50.520.52.50.530.540.5

## Worked example

Inputs — Function: y = A·sin(Bx − C) + D; A (amplitude factor): 2; B (frequency factor): 3; C (phase): 1.5; D (vertical shift): 1; Evaluate at x: 0.

Results — **Phase shift C/B = 0.5**; Amplitude = 2; Period = 2.0943951; Frequency (cycles per unit) = 0.47746483; Vertical shift = 1; Maximum value = 3; Minimum value = -1; y at x = -0.99498997.

## Key terms

- **Amplitude**: Half the distance between the maximum and minimum.
- **Period**: The horizontal length of one full cycle.

## FAQ

**Which direction is a positive phase shift?**
To the right, for the form sin(Bx − C) with C/B > 0.

**Why divide C by B?**
Bx − C = B(x − C/B), so the graph is shifted by C/B, not C.

**What does a negative A do?**
Flips the wave upside down; the amplitude is still |A|.

**What is the period of sin(3x)?**
2π/3 ≈ 2.094.


## Sources

- [OpenStax – Algebra and Trigonometry 2e](https://openstax.org/details/books/algebra-and-trigonometry-2e)
- [OpenStax – Precalculus 2e](https://openstax.org/details/books/precalculus-2e)
- [Wolfram MathWorld – Phase](https://mathworld.wolfram.com/Phase.html)