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

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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.

A General Note Label

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 = \frac{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 $f (x) = \frac{1}{2} \sin (x) ?$ Is the function stretched or compressed vertically?

$\frac{1}{2}$ 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:

\begin{matrix} y = A \sin (B x - C) + D and y = A \cos (B x - C) + D \\ o r \\ y = A \sin (B (x - \frac{C}{B})) + D and y = A \cos (B (x - \frac{C}{B})) + D \end{matrix}

The value $\frac{C}{B}$ 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 $f (x) = \sin (x - \frac{π}{4}),$ which shifts to the right by $\frac{π}{4}$ 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.

A General Note label

Variations of sine and cosine functions

Given an equation in the form $f (x) = A \sin (B x - C) + D$ or $f (x) = A \cos (B x - C) + D,$ $\frac{C}{B}$ 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 $f (x) = \sin (x + \frac{π}{6}) - 2.$

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 $C = - \frac{π}{6} .$ So the phase shift is

\begin{array}{r} \frac{C}{B} = - \frac{\frac{π}{6}}{1} \\ = - \frac{π}{6} \end{array}

or $\frac{π}{6}$ units to the left.

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Determine the direction and magnitude of the phase shift for $f (x) = 3 \cos (x - \frac{π}{2}) .$

$\frac{π}{2};$ 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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How to Feature

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

Determine the amplitude as $| A | .$
Determine the period as $P = \frac{2 π}{| B |} .$
Determine the phase shift as $\frac{C}{B} .$
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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