8.1 Graphs of the sine and cosine functions  (Page 5/13)

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Write a formula for the function graphed in [link] .

two possibilities: $\text{\hspace{0.17em}}y=4\mathrm{sin}\left(\frac{\pi }{5}x-\frac{\pi }{5}\right)+4\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}y=-4\mathrm{sin}\left(\frac{\pi }{5}x+\frac{4\pi }{5}\right)+4$

Graphing variations of y = sin x And y = cos x

Throughout this section, we have learned about types of variations of sine and cosine functions and used that information to write equations from graphs. Now we can use the same information to create graphs from equations.

Instead of focusing on the general form equations

we will let $\text{\hspace{0.17em}}C=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}D=0\text{\hspace{0.17em}}$ and work with a simplified form of the equations in the following examples.

Given the function $\text{\hspace{0.17em}}y=A\mathrm{sin}\left(Bx\right),\text{\hspace{0.17em}}$ sketch its graph.

1. Identify the amplitude, $\text{\hspace{0.17em}}|A|.$
2. Identify the period, $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}.$
3. Start at the origin, with the function increasing to the right if $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is positive or decreasing if $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is negative.
4. At $\text{\hspace{0.17em}}x=\frac{\pi }{2|B|}\text{\hspace{0.17em}}$ there is a local maximum for $\text{\hspace{0.17em}}A>0\text{\hspace{0.17em}}$ or a minimum for $\text{\hspace{0.17em}}A<0,\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}y=A.$
5. The curve returns to the x -axis at $\text{\hspace{0.17em}}x=\frac{\pi }{|B|}.$
6. There is a local minimum for $\text{\hspace{0.17em}}A>0\text{\hspace{0.17em}}$ (maximum for $\text{\hspace{0.17em}}A<0$ ) at $\text{\hspace{0.17em}}x=\frac{3\pi }{2|B|}\text{\hspace{0.17em}}$ with $\text{\hspace{0.17em}}y=–A.$
7. The curve returns again to the x -axis at $\text{\hspace{0.17em}}x=\frac{\pi }{2|B|}.$

Graphing a function and identifying the amplitude and period

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=-2\mathrm{sin}\left(\frac{\pi x}{2}\right).$

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

• Step 1. We can see from the equation that $\text{\hspace{0.17em}}A=-2,$ so the amplitude is 2.
$|A|=2$
• Step 2. The equation shows that $\text{\hspace{0.17em}}B=\frac{\pi }{2},\text{\hspace{0.17em}}$ so the period is
• Step 3. Because $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is negative, the graph descends as we move to the right of the origin.
• Step 4–7. The x -intercepts are at the beginning of one period, $\text{\hspace{0.17em}}x=0,\text{\hspace{0.17em}}$ the horizontal midpoints are at $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ and at the end of one period at $\text{\hspace{0.17em}}x=4.$

The quarter points include the minimum at $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ and the maximum at $\text{\hspace{0.17em}}x=3.\text{\hspace{0.17em}}$ A local minimum will occur 2 units below the midline, at $\text{\hspace{0.17em}}x=1,\text{\hspace{0.17em}}$ and a local maximum will occur at 2 units above the midline, at $\text{\hspace{0.17em}}x=3.\text{\hspace{0.17em}}$ [link] shows the graph of the function.

Sketch a graph of $\text{\hspace{0.17em}}g\left(x\right)=-0.8\mathrm{cos}\left(2x\right).\text{\hspace{0.17em}}$ Determine the midline, amplitude, period, and phase shift.

midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ amplitude: $\text{\hspace{0.17em}}|A|=0.8;\text{\hspace{0.17em}}$ period: $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}=\pi ;\text{\hspace{0.17em}}$ phase shift: $\text{\hspace{0.17em}}\frac{C}{B}=0\text{\hspace{0.17em}}$ or none

Given a sinusoidal function with a phase shift and a vertical shift, sketch its graph.

1. Express the function in the general form
2. Identify the amplitude, $\text{\hspace{0.17em}}|A|.$
3. Identify the period, $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}.$
4. Identify the phase shift, $\text{\hspace{0.17em}}\frac{C}{B}.$
5. Draw the graph of $\text{\hspace{0.17em}}f\left(x\right)=A\mathrm{sin}\left(Bx\right)\text{\hspace{0.17em}}$ shifted to the right or left by $\text{\hspace{0.17em}}\frac{C}{B}\text{\hspace{0.17em}}$ and up or down by $\text{\hspace{0.17em}}D.$

Graphing a transformed sinusoid

Sketch a graph of $\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{sin}\left(\frac{\pi }{4}x-\frac{\pi }{4}\right).$

• Step 1. The function is already written in general form: $\text{\hspace{0.17em}}f\left(x\right)=3\mathrm{sin}\left(\frac{\pi }{4}x-\frac{\pi }{4}\right).$ This graph will have the shape of a sine function    , starting at the midline and increasing to the right.
• Step 2. $\text{\hspace{0.17em}}|A|=|3|=3.\text{\hspace{0.17em}}$ The amplitude is 3.
• Step 3. Since $\text{\hspace{0.17em}}|B|=|\frac{\pi }{4}|=\frac{\pi }{4},\text{\hspace{0.17em}}$ we determine the period as follows.
$P=\frac{2\pi }{|B|}=\frac{2\pi }{\frac{\pi }{4}}=2\pi \cdot \frac{4}{\pi }=8$

The period is 8.

• Step 4. Since $\text{\hspace{0.17em}}C=\frac{\pi }{4},\text{\hspace{0.17em}}$ the phase shift is
$\frac{C}{B}=\frac{\frac{\pi }{4}}{\frac{\pi }{4}}=1.$

The phase shift is 1 unit.

• Step 5. [link] shows the graph of the function.

Draw a graph of $\text{\hspace{0.17em}}g\left(x\right)=-2\mathrm{cos}\left(\frac{\pi }{3}x+\frac{\pi }{6}\right).\text{\hspace{0.17em}}$ Determine the midline, amplitude, period, and phase shift.

midline: $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ amplitude: $\text{\hspace{0.17em}}|A|=2;\text{\hspace{0.17em}}$ period: $\text{\hspace{0.17em}}P=\frac{2\pi }{|B|}=6;\text{\hspace{0.17em}}$ phase shift: $\text{\hspace{0.17em}}\frac{C}{B}=-\frac{1}{2}$

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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