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
| Field | What to enter |
|---|---|
| Function | Choose one: y = A·sin(Bx − C) + D, y = A·cos(Bx − C) + D |
| A (amplitude factor) | A number |
| B (frequency factor) | A number |
| C (phase) | A number |
| D (vertical shift) | A number |
| Evaluate at x | A 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/B |
|---|---|
| 1 | 0.5 |
| 1.5 | 0.5 |
| 2 | 0.5 |
| 2.5 | 0.5 |
| 3 | 0.5 |
| 4 | 0.5 |
Formula
Amplitude = |A|, Period = 2π/|B|, Phase shift = C/B, Vertical shift = D
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.
Frequently asked questions
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.