Phase Shift Calculator
Read every transformation off a sinusoid at once. Enter A, B, C, and D from y = A·sin(Bx − C) + D (or cosine — pick the function) and get the phase shift with direction, amplitude, period, vertical shift, and the factored form that makes the shift visible.
Example: with Function sine — y = A sin(Bx − C) + D · A (amplitude factor) 3 · B (angular frequency) 2 · C (phase constant) 4 · D (vertical shift) 1 → Phase shift C/B: 2 — shifts 2 units right.
- Amplitude |A|3
- Period 2π/|B|3.141593 (= π)
- Vertical shift / midline1 — midline y = 1
Computed by the calculator below using its default values. Change any input to see your own numbers.
For y = A sin(Bx − C) + D: amplitude |A|, period 2π/|B|, phase shift C/B (positive = right), midline y = D. The shift is C/B, not C — factor B out to see it.
Why the shift is C/B, not C
Write y = 3 sin(2x − 4) + 1 and it is tempting to read a shift of 4. But the shift is what x must do to restart the wave, and x is being doubled before C is removed. Factor B out and the truth appears: 2x − 4 = 2(x − 2), so the graph of sin(2x) slides right by 2, not 4. In general y = A sin(Bx − C) + D moves C/B units — right when C/B is positive, left when negative.
The other three constants read off directly. |A| is the amplitude, the height from midline to crest (a negative A flips the graph). B compresses time: the period is 2π/|B|, so B = 2 fits a full wave into π. D lifts the whole curve, planting the midline at y = D.
Same wave, two spellings
Some textbooks write the factored form y = A sin(B(x − h)) + k from the start; there the phase shift is simply h. Both conventions describe identical graphs — C = Bh converts between them. When comparing answers with a classmate or a solutions manual, check which spelling the source uses before concluding someone is wrong. And the units are whatever x is measured in: radians for pure math, seconds for signals, degrees if you built the equation that way.
How it’s calculated
For y = A·sin(Bx − C) + D or A·cos(Bx − C) + D with A ≠ 0, B ≠ 0: amplitude = |A|; period = 2π/|B| (π = 3.14159265); phase shift = C/B, displacing the graph right when positive, left when negative; vertical shift = D (midline y = D). Factored form: y = A f(B(x − C/B)) + D. All in radians; x-axis units follow your input.
Assumes the equation is already in the form A f(Bx − C) + D — if yours is written A f(B(x − h)) + k, the phase shift is h directly and no division is needed.
Transformations of y = sin(x)
| Equation | Amplitude | Period | Phase shift | Midline |
|---|---|---|---|---|
| y = sin(x − π/2) | 1 | 2π | π/2 right | y = 0 |
| y = sin(2x − π) | 1 | π | π/2 right | y = 0 |
| y = 3 sin(x) + 2 | 3 | 2π | none | y = 2 |
| y = sin(x + 1) | 1 | 2π | 1 left | y = 0 |
| y = −2 sin(4x) | 2 (flipped) | π/2 | none | y = 0 |
Computed with amplitude |A|, period 2π/|B|, shift C/B from each equation as written.
Common mistakes
- Reporting C as the shift: in sin(2x − π) the shift is π/2, because the 2 must be factored out first. Divide C by B every time.
- Getting the direction backward — with the Bx − C spelling, a positive C/B moves the graph right. sin(x + 1) has C = −1, so it moves left.
- Using B as the period: B counts how many waves fit in 2π. The period itself is 2π/B — B = 2 means period π, not 2.
- Reading amplitude as A with its sign: amplitude is |A|; the minus in −2 sin(4x) is a reflection, and the wave still rises 2 above the midline.
Frequently asked questions
What is the phase shift formula?
For y = A sin(Bx − C) + D, phase shift = C/B: positive shifts the graph right, negative shifts left. In y = 3 sin(2x − 4) + 1 the shift is 4/2 = 2 units right.
How do I find the amplitude?
Amplitude = |A|, the coefficient in front of sine or cosine, ignoring sign. It is the vertical distance from the midline to a peak. y = −5 sin(x) has amplitude 5; the negative sign only reflects the graph.
What is the period, and how does B change it?
Period = 2π/|B|, the horizontal length of one complete wave. Larger B compresses the wave: B = 2 gives period π, B = ½ stretches it to 4π.
My textbook writes y = A sin(B(x − h)) + k. Is the shift still C/B?
In that factored spelling the shift is just h — the division is already done. The two forms match via C = B·h, so sin(2x − 4) and sin(2(x − 2)) are the same graph shifted 2 right.
Does phase shift work the same for cosine?
Yes — C/B with the same direction rule. Cosine is itself sine shifted left by π/2, which is why cos(x) = sin(x + π/2); the calculator treats both functions identically.