Phase Shift

Phase Shift

Horizontal shift for a periodic function.

For example, the function f(x) = 3sin (x – π) has a phase shift of π. That is, the graph of f(x) = 3sin x is shifted π units to the right.

See also

Translation, sine

Key Formula

y = A\sin\!\bigl(B(x - C)\bigr) + D \quad \text{or} \quad y = A\cos\!\bigl(B(x - C)\bigr) + D

Where:

$A$ = Amplitude — the vertical stretch or compression of the wave
$B$ = Frequency factor — determines the period via Period = 2π / |B|
$C$ = Phase shift — the horizontal shift; positive C shifts the graph right, negative C shifts it left
$D$ = Vertical shift — moves the midline of the wave up or down

Worked Example

Problem: Find the phase shift of y = 2sin(3x − π) + 1.

Step 1: Write the argument in factored form B(x − C). Factor out the coefficient of x from the expression inside the sine.

3x - \pi = 3\!\left(x - \frac{\pi}{3}\right)

Step 2: Identify C by comparing with the standard form sin(B(x − C)). Here, the expression inside the parentheses is x − π/3.

C = \frac{\pi}{3}

Step 3: Determine the direction. Since C = π/3 is positive, the graph shifts to the right.

\text{Phase shift} = \frac{\pi}{3} \text{ units to the right}

Answer: The phase shift is π/3 units to the right.

Another Example

This example shows a leftward phase shift (negative C) and uses cosine instead of sine. It also demonstrates handling a plus sign inside the argument, which students often find confusing.

Problem: Find the phase shift of y = −cos(2x + π/2).

Step 1: The argument of cosine is 2x + π/2. Factor out the coefficient of x.

2x + \frac{\pi}{2} = 2\!\left(x + \frac{\pi}{4}\right)

Step 2: Rewrite in standard form B(x − C). Notice that x + π/4 = x − (−π/4).

2\!\left(x - \left(-\frac{\pi}{4}\right)\right) \implies C = -\frac{\pi}{4}

Step 3: Interpret the sign: C is negative, so the graph shifts to the left.

\text{Phase shift} = \frac{\pi}{4} \text{ units to the left}

Answer: The phase shift is π/4 units to the left.

Frequently Asked Questions

How do you find the phase shift from an equation?

Factor out the coefficient B from the argument of the trig function so it looks like B(x − C). The value C is the phase shift. If C is positive, the shift is to the right; if C is negative, the shift is to the left. A common error is forgetting to factor out B first — you must divide the constant term by B to get C.

What is the difference between phase shift and horizontal shift?

For sine and cosine functions, phase shift and horizontal shift mean the same thing: a left or right translation of the graph. The term "phase shift" is specific to periodic functions and originates from physics and signal processing, while "horizontal shift" is a more general term used for any type of function.

Can the phase shift be negative?

Yes. A negative phase shift (C < 0) means the graph moves to the left. For example, y = sin(x + π) has C = −π, so the graph of sin(x) shifts π units to the left. The magnitude of C gives the distance, and the sign gives the direction.

Phase Shift vs. Vertical Shift

	Phase Shift	Vertical Shift
Definition	Horizontal translation of a periodic function	Vertical translation of the entire graph up or down
Parameter	C in y = A sin(B(x − C)) + D	D in y = A sin(B(x − C)) + D
Effect on graph	Shifts the wave left or right along the x-axis	Shifts the wave up or down along the y-axis (moves the midline)
How to identify	Factor out B from the argument and read off C	Read the constant D added outside the trig function

Why It Matters

Phase shift appears throughout precalculus and trigonometry whenever you graph or analyze sinusoidal functions. In physics, it describes how waves (sound, light, electrical signals) are offset in time relative to each other — for instance, alternating current circuits rely heavily on phase relationships. Understanding phase shift is also essential for modeling real-world periodic data, such as tides or seasonal temperatures, where the cycle does not begin at the standard starting point.

Common Mistakes

Mistake: Reading the constant term directly as the phase shift without factoring out B.

Correction: In y = sin(3x − π), the phase shift is NOT π. You must first factor: 3(x − π/3), so the phase shift is π/3. Always divide the constant inside the argument by B.

Mistake: Getting the direction of the shift backwards when there is a plus sign.

Correction: In y = cos(x + π/4), the argument is x − (−π/4), so C = −π/4. The shift is π/4 to the LEFT, not to the right. Remember: a plus sign inside means a shift to the left.

Related Terms

Horizontal Shift — General term for left/right translation of any function
Periodic Function — Type of function where phase shift applies
Sine — Primary trig function used with phase shift
Amplitude — Vertical stretch parameter A in the same formula
Period — Horizontal length of one cycle, determined by B
Shift — General concept covering both horizontal and vertical shifts
Function — Foundational concept — phase shift modifies a function
Graph of an Equation or Inequality — Visual representation where phase shift is observed