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:

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:

OR

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

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

• 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:

1. b = 2 and c = −π/2​.
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:

1. Compare the equation with the general formula
2. Here, A = 3, B = 4, and C = 0.5
3. 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

Solution:

To find the phase shift:

1. Compare with y = A \sin(Bx - C) + D
2. Here, A = 3, B = 4, and C = -3.
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

Solution:

To find the phase shift:

1. Compare with .
2. Here, A = 2, B = 2, and C = -1.
3. Phase shift formula:

The phase shift is 0.5 units to the right.

Example 4: Find phase shift for function

Solution:

1. Vertical shift D = 4, meaning the graph moves 4 units up.
2. Phase shift formula:

The phase shift is ​ units to the right.

Worksheet: Phase Shift

Problem 1: Find the phase shift:

Problem 2: Find the phase shift:

Problem 3: Find the phase shift:

Problem 4: Find the phase shift:

Problem 5: Find the phase shift:

Problem 6: Find the phase shift:

Problem 7: Find the phase shift:

Problem 8: Find the phase shift:

Answer Key

1. ​ (shifted left by )
2. ​ (shifted right by )
3. ​ (shifted left by ​)
4. (shifted right by ​)
5. (shifted left by )
6. (shifted right by 1 unit)
7. -1 (shifted left by 1 unit)
8. π (shifted right by π units)

Read More,

• Trigonometry
• Trigonometric Identities
• Sine Function
• Cosine Function