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A graph with four items. The x-axis ranges from -6pi to 6pi. The y-axis ranges from -4 to 4. The first item is the graph of sin(x), which has an amplitude of 1. The second is a graph of 2sin(x), which has amplitude of 2. The third is a graph of 3sin(x), which has an amplitude of 3. The fourth is a graph of 4 sin(x) with an amplitude of 4.

Amplitude of sinusoidal functions

If we let C = 0 and D = 0 in the general form equations of the sine and cosine functions, we obtain the forms

y = A sin ( B x )  and  y = A cos ( B x )

The amplitude    is A , and the vertical height from the midline    is | A | . In addition, notice in the example that

| A |  = amplitude =  1 2 | maximum   minimum |

Identifying the amplitude of a sine or cosine function

What is the amplitude of the sinusoidal function f ( x ) = −4 sin ( x ) ? Is the function stretched or compressed vertically?

Let’s begin by comparing the function to the simplified form y = A sin ( B x ) .

In the given function, A = −4 , so the amplitude is | A | = | −4 | = 4. The function is stretched.

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What is the amplitude of the sinusoidal function f ( x ) = 1 2 sin ( x ) ? Is the function stretched or compressed vertically?

1 2 compressed

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Analyzing graphs of variations of y = sin x And y = cos x

Now that we understand how A and B relate to the general form equation for the sine and cosine functions, we will explore the variables C and D . Recall the general form:

y = A sin ( B x C ) + D  and  y = A cos ( B x C ) + D o r y = A sin ( B ( x C B ) ) + D  and  y = A cos ( B ( x C B ) ) + D

The value C B for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function    . If C > 0 , the graph shifts to the right. If C < 0 , the graph shifts to the left. The greater the value of | C | , the more the graph is shifted. [link] shows that the graph of f ( x ) = sin ( x π ) shifts to the right by π units, which is more than we see in the graph of f ( x ) = sin ( x π 4 ) , which shifts to the right by π 4 units.

A graph with three items. The first item is a graph of sin(x). The second item is a graph of sin(x-pi/4), which is the same as sin(x) except shifted to the right by pi/4. The third item is a graph of sin(x-pi), which is the same as sin(x) except shifted to the right by pi.

While C relates to the horizontal shift, D indicates the vertical shift from the midline in the general formula for a sinusoidal function. See [link] . The function y = cos ( x ) + D has its midline at y = D .

A graph of y=Asin(x)+D. Graph shows the midline of the function at y=D.

Any value of D other than zero shifts the graph up or down. [link] compares f ( x ) = sin x with f ( x ) = sin x + 2 , which is shifted 2 units up on a graph.

A graph with two items. The first item is a graph of sin(x). The second item is a graph of sin(x)+2, which is the same as sin(x) except shifted up by 2.

Variations of sine and cosine functions

Given an equation in the form f ( x ) = A sin ( B x C ) + D or f ( x ) = A cos ( B x C ) + D , C B is the phase shift    and D is the vertical shift    .

Identifying the phase shift of a function

Determine the direction and magnitude of the phase shift for f ( x ) = sin ( x + π 6 ) 2.

Let’s begin by comparing the equation to the general form y = A sin ( B x C ) + D .

In the given equation, notice that B = 1 and C = π 6 . So the phase shift is

C B = π 6 1     = π 6

or π 6 units to the left.

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Determine the direction and magnitude of the phase shift for f ( x ) = 3 cos ( x π 2 ) .

π 2 ; right

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Identifying the vertical shift of a function

Determine the direction and magnitude of the vertical shift for f ( x ) = cos ( x ) 3.

Let’s begin by comparing the equation to the general form y = A cos ( B x C ) + D .

In the given equation, D = −3 so the shift is 3 units downward.

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Determine the direction and magnitude of the vertical shift for f ( x ) = 3 sin ( x ) + 2.

2 units up

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Given a sinusoidal function in the form f ( x ) = A sin ( B x C ) + D , identify the midline, amplitude, period, and phase shift.

  1. Determine the amplitude as | A | .
  2. Determine the period as P = 2 π | B | .
  3. Determine the phase shift as C B .
  4. Determine the midline as y = D .

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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