Functions Worksheets

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Phase Shift

By Vighnesh Hemnani

Functions, especially trigonometry functions are not always observed in the standard state like sin(x). In real life, these functions are irregular and need to be transformed into a state that we can better understand. Phase shift is one of the ways that we can transform them. At Thinkster, learn about what they are and how to use them!

Why is this concept useful?

Modelling real life phenomena is a very crucial application of mathematics in the physical world and can help predict a lot of events, however, these movements do not occur in the standard/regular state of sin(x) or cos(x). These observations are complex and irregular with all kinds of transformations. This is why phase shifts are really important to understand how the function measured from a real moving object differs from the standard way we think of sin(x) or cos(x). Did it move to the left or right? Phase shifting is one example of a graph transformation, they are many more that help understand what truly occurs and depict a better picture.

Where does this concept fit into the curriculum?

High School Functions

What is phase shifting?

As per the definition, a phase shift is the distance a function has horizontally moved from the original position of the function (usually (0,0) but the starting point may change based on the original function). This movement can be either left or right and the terminology of “phase shifting” is generally applied to trigonometric functions as they have phases. The generic counter-part term for all functions (such as algebraic or exponential) is just a horizontal shift.

How to use this concept?

One of the key ways to use the ideas of phase shift is to be able to determine the phase shift just from a graph so that we can get a better idea of how the graph changed. For example, from the graph below, we can tell that the function could look like a sinusoidal function, but it is horizontally moved to the right. It is important to note that when a graph is moved the left by x distance, then we add x from the inside of the function. However, if the graph is moved to the right, then we subtract.

(redraw diagram and rewrite red and blue label with arrows)

Based on the above diagram, we see that the blue graph has moved to the left by

π4\frac{\pi}{4}
so the phase shift is
π4\frac{\pi}{4}
and in the equation for the blue function, we subtract the phase shift from x. If it were to be the opposite case of moving to the right, then the phase shift would be
−π4-\frac{\pi}{4}
and we would add
π4\frac{\pi}{4}
to x (essentially we are doing x -(
−π4-\frac{\pi}{4}
) = x + (
π4\frac{\pi}{4}
) because the original form of the equation is:

y=Asin(B(x−C))+D).y = Asin (B(x-C))+D).

Note: A common mistake when finding the phase shift is not ensuring that the function is in the right form. The original form of sinusoidal function is:

y=Asin(B(x−C))+D).y = Asin (B(x-C))+D).

Note how the coefficient of x is 1 and the B is factored out. Now, let us review a common mistake. Let’s say Tiffany says the phase shift is

π2\frac{\pi}{2}
in
KaTeX can only parse string typed expression
y = sin sin( 2x -
π2\frac{\pi}{2}
). Unfortunately, Tiffany would be wrong because the equation is not in the original form. First we need to factor out the 2, to make the coefficient of x equal to 1, then we can say that the equation is:

y=sinsin(2(x−(π4))y= sin sin (2(x-(\frac{\pi}{4}))

So the phase shift is actually

π4\frac{\pi}{4}

Sample Math Problems

Question

Consider the equation y = cos cos ( 3x-2

π\pi
) +2. What would be the phase shift?

Solution

First we would need to convert the equation to the standard form like so:

y=coscos(3(x−2π3))+2y = cos cos (3(x-\frac{2\pi}{3}))+ 2

And now, we can tell that the phase shift should be

2π3\frac{2\pi}{3}

Question

Consider the equation:

y=45sinsin(π2−4x)+3y = \frac{4}{5} sin sin (\frac{\pi}{2} - 4x) + 3

What would be the phase shift?

Solution

First we would need to convert the equation to the standard form like so:

y=45sinsin(−4(x−π8))+3y = \frac{4}{5} sin sin (-4(x - \frac{\pi}{8} )) + 3

And now, we can tell that the phase shift should be

π8\frac{\pi}{8}

Question

The graph below is of a transformed tangent function. What is its phase shift?

(redraw graph – it is taken from a website)

Solution

This is a trick question. There is no phase shift (Phase shift = 0) because the original tangent function is centered at the y-axis, and so is this graph, therefore there has been no horizontal movement.

Question

Consider the equation:

y=coscos(−x−π16)+7y = cos cos( -x - \frac{\pi}{16}) + 7

What would be the phase shift?

Solution

First we would need to convert the equation to the standard form like so:

y=coscos(−(x+π16))+7y = cos cos( -(x + \frac{\pi}{16}) ) + 7

And now, we can tell that the phase shift should be -

π16\frac{\pi}{16}

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Practice Math Problems

1. Consider the equation y = sin sinx +

2π3\frac{2\pi}{3}
. What would be the phase shift?

2. Consider the equation y = sin sin( x +

π\pi
) + 12. What would be the phase shift?

3. Consider the equation y = 2x +

π\pi
.What would be the phase shift?

4. Consider the equation y = 10coscos 10x - 10

π\pi
. What would be the phase shift?

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