# 8.1 Graphs of the sine and cosine functions  (Page 3/13)

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## Amplitude of sinusoidal functions

If we let $\text{\hspace{0.17em}}C=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}D=0\text{\hspace{0.17em}}$ in the general form equations of the sine and cosine functions, we obtain the forms

The amplitude    is $\text{\hspace{0.17em}}A,\text{\hspace{0.17em}}$ and the vertical height from the midline    is $\text{\hspace{0.17em}}|A|.\text{\hspace{0.17em}}$ In addition, notice in the example that

## Identifying the amplitude of a sine or cosine function

What is the amplitude of the sinusoidal function $\text{\hspace{0.17em}}f\left(x\right)=-4\mathrm{sin}\left(x\right)?\text{\hspace{0.17em}}$ Is the function stretched or compressed vertically?

Let’s begin by comparing the function to the simplified form $\text{\hspace{0.17em}}y=A\mathrm{sin}\left(Bx\right).$

In the given function, $\text{\hspace{0.17em}}A=-4,\text{\hspace{0.17em}}$ so the amplitude is $\text{\hspace{0.17em}}|A|=|-4|=4.\text{\hspace{0.17em}}$ The function is stretched.

What is the amplitude of the sinusoidal function $\text{\hspace{0.17em}}f\left(x\right)=\frac{1}{2}\mathrm{sin}\left(x\right)?\text{\hspace{0.17em}}$ Is the function stretched or compressed vertically?

$\frac{1}{2}\text{\hspace{0.17em}}$ compressed

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

Now that we understand how $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}B\text{\hspace{0.17em}}$ relate to the general form equation for the sine and cosine functions, we will explore the variables $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}D.\text{\hspace{0.17em}}$ Recall the general form:

The value $\text{\hspace{0.17em}}\frac{C}{B}\text{\hspace{0.17em}}$ for a sinusoidal function is called the phase shift , or the horizontal displacement of the basic sine or cosine function    . If $\text{\hspace{0.17em}}C>0,\text{\hspace{0.17em}}$ the graph shifts to the right. If $\text{\hspace{0.17em}}C<0,\text{\hspace{0.17em}}$ the graph shifts to the left. The greater the value of $\text{\hspace{0.17em}}|C|,\text{\hspace{0.17em}}$ the more the graph is shifted. [link] shows that the graph of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(x-\pi \right)\text{\hspace{0.17em}}$ shifts to the right by $\text{\hspace{0.17em}}\pi \text{\hspace{0.17em}}$ units, which is more than we see in the graph of $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(x-\frac{\pi }{4}\right),\text{\hspace{0.17em}}$ which shifts to the right by $\text{\hspace{0.17em}}\frac{\pi }{4}\text{\hspace{0.17em}}$ units.

While $\text{\hspace{0.17em}}C\text{\hspace{0.17em}}$ relates to the horizontal shift, $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ indicates the vertical shift from the midline in the general formula for a sinusoidal function. See [link] . The function $\text{\hspace{0.17em}}y=\mathrm{cos}\left(x\right)+D\text{\hspace{0.17em}}$ has its midline at $\text{\hspace{0.17em}}y=D.$

Any value of $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ other than zero shifts the graph up or down. [link] compares $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\text{\hspace{0.17em}}x+2,\text{\hspace{0.17em}}$ which is shifted 2 units up on a graph.

## Variations of sine and cosine functions

Given an equation in the form $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{sin}\left(Bx-C\right)+D\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{cos}\left(Bx-C\right)+D,\text{\hspace{0.17em}}$ $\frac{C}{B}\text{\hspace{0.17em}}$ is the phase shift    and $\text{\hspace{0.17em}}D\text{\hspace{0.17em}}$ is the vertical shift    .

## Identifying the phase shift of a function

Determine the direction and magnitude of the phase shift for $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{sin}\left(x+\frac{\pi }{6}\right)-2.$

Let’s begin by comparing the equation to the general form $\text{\hspace{0.17em}}y=A\mathrm{sin}\left(Bx-C\right)+D.$

In the given equation, notice that $\text{\hspace{0.17em}}B=1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}C=-\frac{\pi }{6}.\text{\hspace{0.17em}}$ So the phase shift is

or $\text{\hspace{0.17em}}\frac{\pi }{6}\text{\hspace{0.17em}}$ units to the left.

Determine the direction and magnitude of the phase shift for $\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{cos}\left(x-\frac{\pi }{2}\right).$

$\frac{\pi }{2};\text{\hspace{0.17em}}$ right

## Identifying the vertical shift of a function

Determine the direction and magnitude of the vertical shift for $\text{\hspace{0.17em}}f\left(x\right)=\mathrm{cos}\left(x\right)-3.$

Let’s begin by comparing the equation to the general form $\text{\hspace{0.17em}}y=A\mathrm{cos}\left(Bx-C\right)+D.$

In the given equation, $\text{\hspace{0.17em}}D=-3\text{\hspace{0.17em}}$ so the shift is 3 units downward.

Determine the direction and magnitude of the vertical shift for $\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{sin}\left(x\right)+2.$

2 units up

Given a sinusoidal function in the form $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{sin}\left(Bx-C\right)+D,\text{\hspace{0.17em}}$ identify the midline, amplitude, period, and phase shift.

1. Determine the amplitude as $\text{\hspace{0.17em}}|A|.$
2. Determine the period as $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}.$
3. Determine the phase shift as $\text{\hspace{0.17em}}\frac{C}{B}.$
4. Determine the midline as $\text{\hspace{0.17em}}y=D.$

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