Phase Shift Formula

Phase shift refers to the horizontal shift or displacement of a wave or periodic function, such as a sine or cosine wave, along the x-axis. In simple words, phase shift is when a function (or the graph of that function) is moved horizontally to the left or right along the x- axis. It is also called a horizontal shift.

The general formula for any transformation of a function can be written as:

y = a \cdot f(bx - c) + d

In this equation, each variable represents a different type of transformation, but for phase shift, we focus on the b and c values. The phase shift is determined by the expression inside the parentheses, bx - c, and tells us how far the graph is shifted horizontally.

Equation for Phase Shift

The general form of the sine and cosine functions incorporating phase shift is as follows:

y = A \sin(B(x + C)) + D OR y = A \sin(Bx - C) + D

Where:

A is the amplitude (the height of the wave from the centerline).
B affects the period of the wave (how often the wave repeats).
C/B represents the phase shift (horizontal shift).
D represents the vertical shift.

Phase Shift Formula helps determine how much a function is shifted horizontally from its original position. Phase shift refers to moving the graph of a function left or right along the x-axis. The Formula for phase shift is given by

\text{Phase Shift} = \frac{C}{B}

A positive phase shift means the graph shifts to the left.
A negative phase shift means the graph shifts to the right.

Example:

For y = 3 ⋅ sin⁡(2x − π/2) + 1:

b = 2 and c = −π/2.
Substitute into the formula: Phase Shift = (−π/2)/2 = π/4

So, the phase shift is π4\frac{\pi}{4}4π units to the right.

Solved Examples on Phase Shift

Example 1: Find phase shift for function F(x) = 3 \sin(4(x - 0.5)) + 2

Solution:

To find the phase shift:
Compare the equation with the general formula y = A \sin(B(x - C)) + D
Here, A = 3, B = 4, and C = 0.5
Use the phase shift formula:
\text{Phase Shift} = \frac{C}{B} = \frac{0.5}{4} = 0.125
The phase shift is 0.125 units to the right.

Example 2: Find phase shift for function F(x) = 3 \sin(4x + 3)

Solution:

To find the phase shift:
Compare with y = A \sin(Bx - C) + Dy = A \sin(Bx - C) + D
Here, A = 3, B = 4, and C = -3.
Phase shift formula:
\text{Phase Shift} = \frac{C}{B} = \frac{-3}{4} = -0.75
The phase shift is 0.75 units to the left.

Example 3: Find phase shift for function F(x) = 2 \cos(2(x + 1)) - 3

Solution:

To find the phase shift:
Compare with y = A \cos(B(x - C)) + D.
Here, A = 2, B = 2, and C = -1.
Phase shift formula:
\text{Phase Shift} = \frac{C}{B} = \frac{-1}{2} = -0.5
The phase shift is 0.5 units to the right.

Example 4: Find phase shift for function F(x) = 5 \sin(3(x - \pi)) + 4

Solution:

Vertical shift D = 4, meaning the graph moves 4 units up.
Phase shift formula:
\text{Phase Shift} = \frac{C}{B} = \frac{\pi}{3}
The phase shift is \frac{\pi}{3} units to the right.

Worksheet: Phase Shift

Problem 1: Find the phase shift: F(x) = 2 \sin(3(x + 2)) - 11

Problem 2: Find the phase shift: F(x) = 4 \cos(5x - 3)

Problem 3: Find the phase shift: F(x) = 3 \sin(6x + \pi)

Problem 4: Find the phase shift: F(x) = \tan(2(x - \frac{\pi}{6}))

Problem 5: Find the phase shift: F(x) = \cos(4x + \frac{\pi}{2})

Problem 6: Find the phase shift: F(x) = 7 \sin(2(x - 1)) + 2

Problem 7: Find the phase shift: F(x) = 5 \cos(3x + 3)

Problem 8: Find the phase shift: F(x) = 4 \sin(x - \pi)

Answer Key

-\frac{2}{3} (shifted left by \frac{2}{3})
\frac{3}{5} (shifted right by \frac{3}{5})
-\frac{\pi}{6} (shifted left by \frac{\pi}{6})
\frac{\pi}{12} (shifted right by \frac{\pi}{12})
-\frac{\pi}{8} (shifted left by \frac{\pi}{8})
\frac{1}{2} (shifted right by 1 unit)
-1 (shifted left by 1 unit)
π (shifted right by π units)

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Phase Shift Formula

Equation for Phase Shift