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Mathematics LibreTexts

6-1. Graphs of the Sine and Cosine Functions

Graphs of the Sine and Cosine Functions
In this section, you will:
  • Graph variations of  y=sin( x )  and  y=cos( x ).
  • Use phase shifts of sine and cosine curves.
<figure class="small" id="Figure_06_01_001" style="color: rgb(0, 0, 0); font-family: 'Times New Roman'; font-size: medium; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: auto; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; widows: 1; word-spacing: 0px; -webkit-text-stroke-width: 0px;"> <figcaption>Light can be separated into colors because of its wavelike properties. (credit: "wonderferret"/ Flickr)</figcaption> A photo of a rainbow colored beam of light stretching across the floor.</figure>

White light, such as the light from the sun, is not actually white at all. Instead, it is a composition of all the colors of the rainbow in the form of waves. The individual colors can be seen only when white light passes through an optical prism that separates the waves according to their wavelengths to form a rainbow.

Light waves can be represented graphically by the sine function. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. In this section, we will interpret and create graphs of sine and cosine functions.

Graphing Sine and Cosine Functions

Recall that the sine and cosine functions relate real number values to the x- and y-coordinates of a point on the unit circle. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. We can create a table of values and use them to sketch a graph. [link] lists some of the values for the sine function on a unit circle.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>x</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0</mn></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 6 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>3</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 6 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>π</mi></annotation-xml></semantics></math>
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0</mn></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>1</mn></annotation-xml></semantics></math> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0</mn></annotation-xml></semantics></math>

Plotting the points from the table and continuing along the x-axis gives the shape of the sine function. See [link].

<figure class="small" id="Figure_06_01_002"> <figcaption>The sine function</figcaption> A graph of sin(x). Local maximum at (pi/2, 1). Local minimum at (3pi/2, -1). Period of 2pi.</figure>

Notice how the sine values are positive between 0 and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>which correspond to the values of the sine function in quadrants I and II on the unit circle, and the sine values are negative between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>which correspond to the values of the sine function in quadrants III and IV on the unit circle. See [link].

<figure class="small" id="Figure_06_01_003"> <figcaption>Plotting values of the sine function</figcaption> A side-by-side graph of a unit circle and a graph of sin(x). The two graphs show the equivalence of the coordinates.</figure>

Now let’s take a similar look at the cosine function. Again, we can create a table of values and use them to sketch a graph.[link] lists some of the values for the cosine function on a unit circle.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi mathvariant="bold">x</mi></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0</mn></annotation-xml></semantics></math> <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 6 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>2</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 3 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>3</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 4 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>5</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 6 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mi>π</mi></annotation-xml></semantics></math>
<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi mathvariant="bold">cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x ) <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>1</mn></annotation-xml></semantics></math> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>3</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mn>0</mn></annotation-xml></semantics></math> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mo>−</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

As with the sine function, we can plots points to create a graph of the cosine function as in [link].

<figure class="medium" id="Figure_06_01_004"> <figcaption>The cosine function</figcaption> A graph of cos(x). Local maxima at (0,1) and (2pi, 1). Local minimum at (pi, -1). Period of 2pi.</figure>

Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −1,1 ].

In both graphs, the shape of the graph repeats after<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>which means the functions are periodic with a period of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>A periodic function is a function for which a specific horizontal shift, P, results in a function equal to the original function:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x+P )=f( x ) for all values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the domain of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>When this occurs, we call the smallest such horizontal shift with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>></mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>theperiod of the function. [link] shows several periods of the sine and cosine functions.

<figure class="small" id="Figure_06_01_005">Side-by-side graphs of sin(x) and cos(x). Graphs show period lengths for both functions, which is 2pi.</figure>

Looking again at the sine and cosine functions on a domain centered at the y-axis helps reveal symmetries. As we can see in[link], the sine function is symmetric about the origin. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>−</mi><mi>sin</mi><mtext> </mtext><mi>x</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> Now we can clearly see this property from the graph.

<figure class="small" id="Figure_06_01_006"> <figcaption>Odd symmetry of the sine function</figcaption> A graph of sin(x) that shows that sin(x) is an odd function due to the odd symmetry of the graph.</figure>

[link] shows that the cosine function is symmetric about the y-axis. Again, we determined that the cosine function is an even function. Now we can see from the graph that <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>cos</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_06_01_007"> <figcaption>Even symmetry of the cosine function</figcaption> A graph of cos(x) that shows that cos(x) is an even function due to the even symmetry of the graph.</figure>
Characteristics of Sine and Cosine Functions

The sine and cosine functions have several distinct characteristics:

  • They are periodic functions with a period of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  • The domain of each function is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> −∞,∞ ) and the range is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −1,1 ].
  • The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is symmetric about the origin, because it is an odd function.
  • The graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is symmetric about the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext>-</mtext></mrow></annotation-xml></semantics></math>axis, because it is an even function.

Investigating Sinusoidal Functions

As we can see, sine and cosine functions have a regular period and range. If we watch ocean waves or ripples on a pond, we will see that they resemble the sine or cosine functions. However, they are not necessarily identical. Some are taller or longer than others. A function that has the same general shape as a sine or cosine function is known as a sinusoidal function. The general forms of sinusoidal functions are

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> Bx−C )+D               and y=Acos( Bx−C )+D

Determining the Period of Sinusoidal Functions

Looking at the forms of sinusoidal functions, we can see that they are transformations of the sine and cosine functions. We can use what we know about transformations to determine the period.

In the general formula,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is related to the period by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | . If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> B |>1, then the period is less than<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and the function undergoes a horizontal compression, whereas if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> B |<1, then the period is greater than<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and the function undergoes a horizontal stretch. For example,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x ), <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mo>=</mo><mn>1,</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>so the period is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mi>,</mi><mtext/></mrow></annotation-xml></semantics></math>which we knew. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2x ), then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>=</mo><mn>2,</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>so the period is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and the graph is compressed. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 2 ), then<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 , so the period is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and the graph is stretched. Notice in [link] how the period is indirectly related to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> B |.

<figure class="small" id="Figure_06_01_008">A graph with three items. The x-axis ranges from 0 to 2pi. The y-axis ranges from -1 to 1. The first item is the graph of sin(x) for one full period. The second is the graph of sin(2x) over two periods. The third is the graph of sin(x/2) for one half of a period.</figure>
Period of Sinusoidal Functions

If we let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>in the general form equations of the sine and cosine functions, we obtain the forms

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> Bx )
<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> Bx )

The period is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | .

Identifying the Period of a Sine or Cosine Function

Determine the period of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=sin( π 6 x ).

Let’s begin by comparing the equation to the general form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

In the given equation,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 6 , so the period will be

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mtable columnalign="left"><mtr><mtd><mrow/></mtd></mtr><mtr><mtd><mi>P</mi><mo>=</mo><mfrac/></mtd></mtr></mtable></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2π |B|   = 2π π 6   =2π⋅ 6 π   =12

Determine the period of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x 3 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>6</mn><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>

Determining Amplitude

Returning to the general formula for a sinusoidal function, we have analyzed how the variable<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>relates to the period. Now let’s turn to the variable<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>so we can analyze how it is related to the amplitude, or greatest distance from rest.<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>represents the vertical stretch factor, and its absolute value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A | is the amplitude. The local maxima will be a distance<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A | above the vertical midline of the graph, which is the line<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>D</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>in this case, the midline is the x-axis. The local minima will be the same distance below the midline. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |>1, the function is stretched. For example, the amplitude of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is twice the amplitude of<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>x</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |<1, the function is compressed. [link] compares several sine functions with different amplitudes.

<figure id="Figure_06_01_009">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.</figure>
Amplitude of Sinusoidal Functions

If we let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>in the general form equations of the sine and cosine functions, we obtain the forms

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> Bx ) and y=Acos( Bx )

The amplitude is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and the vertical height from the midline is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |. In addition, notice in the example that

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A | = amplitude =  1 2 | maximum − minimum |
Identifying the Amplitude of a Sine or Cosine Function

What is the amplitude of the sinusoidal function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>−4</mn><mi>sin</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>?</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Is the function stretched or compressed vertically?

Let’s begin by comparing the function to the simplified form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

In the given function,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>=</mo><mn>−4</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>so the amplitude is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |=| −4 |=4. The function is stretched.

Analysis

The negative value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>results in a reflection across the x-axis of the sine function, as shown in [link].

<figure class="small" id="Figure_06_01_010">A graph of -4sin(x). The function has an amplitude of 4. Local minima at (-3pi/2, -4) and (pi/2, -4). Local maxima at (-pi/2, 4) and (3pi/2, 4). Period of 2pi.</figure>

What is the amplitude of the sinusoidal function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 sin(x)? Is the function stretched or compressed vertically?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mn>1</mn></mfrac></mrow></annotation-xml></semantics></math> 2  compressed

Analyzing Graphs of Variations of y = sin x and y = cos x

Now that we understand how<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>relate to the general form equation for the sine and cosine functions, we will explore the variables<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Recall the general form:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable><mtr><mtd><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> Bx−C )+D and y=Acos( Bx−C )+D or y=Asin( B( x− C B ) )+D and y=Acos( B( x− C B ) )+D

The value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B  for a sinusoidal function is called the phase shift, or the horizontal displacement of the basic sine or cosine function. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>></mo><mn>0</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the graph shifts to the right. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo><</mo><mn>0</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the graph shifts to the left. The greater the value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> C |, the more the graph is shifted. [link] shows that the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x−π ) shifts to the right by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>units, which is more than we see in the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x− π 4 ), which shifts to the right by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4  units.

<figure class="small" id="Figure_06_01_011">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.</figure>

While<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>relates to the horizontal shift,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>indicates the vertical shift from the midline in the general formula for a sinusoidal function. See [link]. The function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )+D has its midline at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>D</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_06_01_012">A graph of y=Asin(x)+D. Graph shows the midline of the function at y=D.</figure>

Any value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>other than zero shifts the graph up or down. [link] compares<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mo>+</mo><mn>2</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>which is shifted 2 units up on a graph.

<figure class="small" id="Figure_06_01_013">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.</figure>
Variations of Sine and Cosine Functions

Given an equation in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=Asin( Bx−C )+D or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=Acos( Bx−C )+D, <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>C</mi></mfrac></mrow></annotation-xml></semantics></math> B  is the phase shift and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is the vertical shift.

Identifying the Phase Shift of a Function

Determine the direction and magnitude of the phase shift for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x+ π 6 )−2.

Let’s begin by comparing the equation to the general form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo>−</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

In the given equation, notice that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> π 6 . So the phase shift is

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="right"><mtr columnalign="right"><mtd columnalign="right"><mrow/></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mfrac><mi>C</mi></mfrac></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> B =− π 6 1    =− π 6

or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 6  units to the left.

Analysis

We must pay attention to the sign in the equation for the general form of a sinusoidal function. The equation shows a minus sign before<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Therefore<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x+ π 6 )−2 can be rewritten as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x−( − π 6 ) )−2. If the value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is negative, the shift is to the left.

Determine the direction and magnitude of the phase shift for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x− π 2 ).

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 2 ; right

Identifying the Vertical Shift of a Function

Determine the direction and magnitude of the vertical shift for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )−3.

Let’s begin by comparing the equation to the general form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mi>cos</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo>−</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

In the given equation,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>=</mo><mn>−3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>so the shift is 3 units downward.

Determine the direction and magnitude of the vertical shift for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )+2.

2 units up

Given a sinusoidal function in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=Asin( Bx−C )+D, identify the midline, amplitude, period, and phase shift.

  1. Determine the amplitude as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |.
  2. Determine the period as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | .
  3. Determine the phase shift as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B .
  4. Determine the midline as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>D</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
Identifying the Variations of a Sinusoidal Function from an Equation

Determine the midline, amplitude, period, and phase shift of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>3</mn><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mn>1.</mn></mrow></annotation-xml></semantics></math>

Let’s begin by comparing the equation to the general form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo>−</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>A</mi><mo>=</mo><mn>3</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>so the amplitude is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |=3.

Next,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>=</mo><mn>2</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>so the period is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | = 2π 2 =π.

There is no added constant inside the parentheses, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and the phase shift is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B = 0 2 =0.

Finally,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>so the midline is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>1.</mn></mrow></annotation-xml></semantics></math>

Analysis

Inspecting the graph, we can determine that the period is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the midline is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and the amplitude is 3. See [link].

<figure class="small" id="Figure_06_01_014">A graph of y=3sin(2x)+1. The graph has an amplitude of 3. There is a midline at y=1. There is a period of pi. Local maximum at (pi/4, 4) and local minimum at (3pi/4, -2).</figure>

Determine the midline, amplitude, period, and phase shift of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 cos( x 3 − π 3 ).

midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>amplitude:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |= 1 2 ; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | =6π; phase shift:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B =π

Identifying the Equation for a Sinusoidal Function from a Graph

Determine the formula for the cosine function in [link].

<figure class="small" id="Figure_06_01_015">A graph of -0.5cos(x)+0.5. The graph has an amplitude of 0.5. The graph has a period of 2pi. The graph has a range of [0, 1]. The graph is also reflected about the x-axis from the parent function cos(x).</figure>

To determine the equation, we need to identify each value in the general form of a sinusoidal function.

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo>−</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> y=Acos(Bx−C)+D

The graph could represent either a sine or a cosine function that is shifted and/or reflected. When<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the graph has an extreme point,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 0,0 ). Since the cosine function has an extreme point for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>let us write our equation in terms of a cosine function.

Let’s start with the midline. We can see that the graph rises and falls an equal distance above and below<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0.5.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>This value, which is the midline, is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the equation, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>=</mo><mn>0.5.</mn></mrow></annotation-xml></semantics></math>

The greatest distance above and below the midline is the amplitude. The maxima are 0.5 units above the midline and the minima are 0.5 units below the midline. So<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |=0.5. Another way we could have determined the amplitude is by recognizing that the difference between the height of local maxima and minima is 1, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |= 1 2 =0.5. Also, the graph is reflected about the x-axis so that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>=</mo><mo>−</mo><mn>0.5.</mn></mrow></annotation-xml></semantics></math>

The graph is not horizontally stretched or compressed, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>=</mo><mn>1;</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and the graph is not shifted horizontally, so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>=</mo><mn>0.</mn></mrow></annotation-xml></semantics></math>

Putting this all together,

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>g</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=−0.5cos( x )+0.5

Determine the formula for the sine function in [link].

<figure class="small" id="Figure_06_01_016">A graph of sin(x)+2. Period of 2pi, amplitude of 1, and range of [1, 3].</figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=sin(x)+2

Identifying the Equation for a Sinusoidal Function from a Graph

Determine the equation for the sinusoidal function in [link].

<figure class="medium" id="Figure_06_01_017">A graph of 3cos(pi/3x-pi/3)-2. Graph has amplitude of 3, period of 6, range of [-5,1].</figure>

With the highest value at 1 and the lowest value at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>−5</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the midline will be halfway between at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>−2.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>So<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>=</mo><mn>−2.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>

The distance from the midline to the highest or lowest value gives an amplitude of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |=3.

The period of the graph is 6, which can be measured from the peak at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>to the next peak at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>7</mn><mo>,</mo></mrow></annotation-xml></semantics></math>or from the distance between the lowest points. Therefore,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | =6. Using the positive value for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>,</mo></mrow></annotation-xml></semantics></math>we find that

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>B</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π P = 2π 6 = π 3

So far, our equation is either<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>3</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 x−C )−2 or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>3</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 x−C )−2. For the shape and shift, we have more than one option. We could write this as any one of the following:

  • a cosine shifted to the right
  • a negative cosine shifted to the left
  • a sine shifted to the left
  • a negative sine shifted to the right

While any of these would be correct, the cosine shifts are easier to work with than the sine shifts in this case because they involve integer values. So our function becomes

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>3</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 x− π 3 )−2 or y=−3cos( π 3 x+ 2π 3 )−2

Again, these functions are equivalent, so both yield the same graph.

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

<figure class="medium" id="Figure_06_01_018">A graph of 4sin((pi/5)x-pi/5)+4. Graph has period of 10, amplitude of 4, range of [0,8].</figure>

two possibilities:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>4</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 5 x− π 5 )+4 or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 5 x+ 4π 5 )+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

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> Bx−C )+D and y=Acos( Bx−C )+D,

we will let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and work with a simplified form of the equations in the following examples.

Given the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> Bx ), sketch its graph.

  1. Identify the amplitude,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |.
  2. Identify the period,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | .
  3. Start at the origin, with the function increasing to the right if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is positive or decreasing if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is negative.
  4. At<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2| B |  there is a local maximum for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>></mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>or a minimum for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo><</mo><mn>0</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  5. The curve returns to the x-axis at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π | B | .
  6. There is a local minimum for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>></mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>(maximum for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo><</mo><mn>0</mn></mrow></annotation-xml></semantics></math>) at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3π 2| B |  with<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mo>–</mo><mi>A</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  7. The curve returns again to the x-axis at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2| B | .
Graphing a Function and Identifying the Amplitude and Period

Sketch a graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=−2sin( πx 2 ).

Let’s begin by comparing the equation to the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo></mrow></annotation-xml></semantics></math>

  • Step 1. We can see from the equation that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>=</mo><mo>−</mo><mn>2</mn><mi>,</mi></mrow></annotation-xml></semantics></math>so the amplitude is 2.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |=2
  • Step 2. The equation shows that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2 , so the period is
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><mi>P</mi><mo>=</mo><mfrac/></mrow></mtd></mtr></mtable></mrow></annotation-xml></semantics></math> 2π π 2   =2π⋅ 2 π   =4
  • Step 3. Because<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>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,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the horizontal midpoints are at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and at the end of one period at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>4.</mn></mrow></annotation-xml></semantics></math>

The quarter points include the minimum at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and the maximum at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>3.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>A local minimum will occur 2 units below the midline, at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>and a local maximum will occur at 2 units above the midline, at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>3.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>[link] shows the graph of the function.

<figure class="small" id="Figure_06_01_019">A graph of -2sin((pi/2)x). Graph has range of [-2,2], period of 4, and amplitude of 2.</figure>

Sketch a graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>g</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=−0.8cos( 2x ). Determine the midline, amplitude, period, and phase shift.

A graph of -0.8cos(2x). Graph has range of [-0.8, 0.8], period of pi, amplitude of 0.8, and is reflected about the x-axis compared to it's parent function cos(x).

midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>amplitude:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |=0.8; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | =π; phase shift:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B =0  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<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mi>sin</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo>−</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi><mtext> or </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mi>cos</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo>−</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  2. Identify the amplitude,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |.
  3. Identify the period,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | .
  4. Identify the phase shift,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B .
  5. Draw the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=Asin( Bx )  shifted to the right or left by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B  and up or down by<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
Graphing a Transformed Sinusoid

Sketch a graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=3sin( π 4 x− π 4 ).

  • Step 1. The function is already written in general form:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 4 x− π 4 ).This graph will have the shape of a sine function, starting at the midline and increasing to the right.
  • Step 2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |=| 3 |=3. The amplitude is 3.
  • Step 3. Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> B |=| π 4 |= π 4 , we determine the period as follows.
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | = 2π π 4 =2π⋅ 4 π =8

    The period is 8.

  • Step 4. Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 4 , the phase shift is
    <math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>C</mi></mfrac></mrow></annotation-xml></semantics></math> B = π 4 π 4 =1.

    The phase shift is 1 unit.

  • Step 5. [link] shows the graph of the function. <figure class="small" id="Figure_06_01_021"> <figcaption>A horizontally compressed, vertically stretched, and horizontally shifted sinusoid</figcaption> A graph of 3sin(*(pi/4)x-pi/4). Graph has amplitude of 3, period of 8, and a phase shift of 1 to the right.</figure>

Draw a graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>g</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>2</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 3 x+ π 6 ). Determine the midline, amplitude, period, and phase shift.

A graph of -2cos((pi/3)x+(pi/6)). Graph has amplitude of 2, period of 6, and has a phase shift of 0.5 to the left.

midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>amplitude:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |=2; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | =6; phase shift:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B =− 1 2

Identifying the Properties of a Sinusoidal Function

Given<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mo>−</mo><mn>2</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 x+π )+3, determine the amplitude, period, phase shift, and horizontal shift. Then graph the function.

Begin by comparing the equation to the general form and use the steps outlined in [link].

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mi>A</mi><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> Bx−C )+D
  • Step 1. The function is already written in general form.
  • Step 2. Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>=</mo><mo>−</mo><mn>2</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the amplitude is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |=2.
  • Step 3. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> B |= π 2 , so the period is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | = 2π π 2 =2π⋅ 2 π =4. The period is 4.
  • Step 4. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>C</mi><mo>=</mo><mo>−</mo><mi>π</mi><mo>,</mo></mrow></annotation-xml></semantics></math>so we calculate the phase shift as<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B = −π, π 2 =−π⋅ 2 π =−2. The phase shift is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>2.</mn></mrow></annotation-xml></semantics></math>
  • Step 5. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>D</mi><mo>=</mo><mn>3</mn><mo>,</mo></mrow></annotation-xml></semantics></math>so the midline is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>3</mn><mo>, </mo></mrow></annotation-xml></semantics></math>and the vertical shift is up 3.

Since<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is negative, the graph of the cosine function has been reflected about the x-axis.

[link] shows one cycle of the graph of the function.

<figure class="small" id="Figure_06_01_028">A graph of -2cos((pi/2)x+pi)+3. Graph shows an amplitude of 2, midline at y=3, and a period of 4.</figure>

Using Transformations of Sine and Cosine Functions

We can use the transformations of sine and cosine functions in numerous applications. As mentioned at the beginning of the chapter, circular motion can be modeled using either the sine or cosine function.

Finding the Vertical Component of Circular Motion

A point rotates around a circle of radius 3 centered at the origin. Sketch a graph of the y-coordinate of the point as a function of the angle of rotation.

Recall that, for a point on a circle of radius r, the y-coordinate of the point is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>r</mi><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> so in this case, we get the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> The constant 3 causes a vertical stretch of the y-values of the function by a factor of 3, which we can see in the graph in [link].

<figure class="small" id="Figure_06_01_023">A graph of 3sin(x). Graph has period of 2pi, amplitude of 3, and range of [-3,3].</figure>
Analysis

Notice that the period of the function is still<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>as we travel around the circle, we return to the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 3,0 ) for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>2</mn><mi>π</mi><mo>,</mo><mn>4</mn><mi>π</mi><mo>,</mo><mn>6</mn><mi>π</mi><mo>,</mo><mn>...</mn><mo>.</mo></mrow></annotation-xml></semantics></math>Because the outputs of the graph will now oscillate between<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>–</mo><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3</mn><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>the amplitude of the sine wave is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>3.</mn></mrow></annotation-xml></semantics></math>

What is the amplitude of the function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>7</mn><mi>cos</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>?</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Sketch a graph of this function.

7

A graph of 7cos(x). Graph has amplitude of 7, period of 2pi, and range of [-7,7].
Finding the Vertical Component of Circular Motion

A circle with radius 3 ft is mounted with its center 4 ft off the ground. The point closest to the ground is labeled P, as shown in [link]. Sketch a graph of the height above the ground of the point<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>as the circle is rotated; then find a function that gives the height in terms of the angle of rotation.

<figure class="small" id="Figure_06_01_025">An illustration of a circle lifted 4 feet off the ground. Circle has radius of 3 ft. There is a point P labeled on the circle's circumference.</figure>

Sketching the height, we note that it will start 1 ft above the ground, then increase up to 7 ft above the ground, and continue to oscillate 3 ft above and below the center value of 4 ft, as shown in [link].

<figure class="small" id="Figure_06_01_026">A graph of -3cox(x)+4. Graph has midline at y=4, amplitude of 3, and period of 2pi.</figure>

Although we could use a transformation of either the sine or cosine function, we start by looking for characteristics that would make one function easier to use than the other. Let’s use a cosine function because it starts at the highest or lowest value, while a sine function starts at the middle value. A standard cosine starts at the highest value, and this graph starts at the lowest value, so we need to incorporate a vertical reflection.

Second, we see that the graph oscillates 3 above and below the center, while a basic cosine has an amplitude of 1, so this graph has been vertically stretched by 3, as in the last example.

Finally, to move the center of the circle up to a height of 4, the graph has been vertically shifted up by 4. Putting these transformations together, we find that

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )+4

A weight is attached to a spring that is then hung from a board, as shown in [link]. As the spring oscillates up and down, the position<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>of the weight relative to the board ranges from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>–1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>in. (at time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>–7</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>in. (at time<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>π</mi><mo stretchy="false">)</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>below the board. Assume the position of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is given as a sinusoidal function of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>Sketch a graph of the function, and then find a cosine function that gives the position<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in terms of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<figure class="small" id="Figure_06_01_029">An illustration of a spring with length y.</figure>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>3</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )−4

A cosine graph with range [-1,-7]. Period is 2 pi. Local maximums at (0,-1), (2pi,-1), and (4pi, -1). Local minimums at (pi,-7) and (3pi, -7).
Determining a Rider’s Height on a Ferris Wheel

The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). It completes one rotation every 30 minutes. Riders board from a platform 2 meters above the ground. Express a rider’s height above ground as a function of time in minutes.

With a diameter of 135 m, the wheel has a radius of 67.5 m. The height will oscillate with amplitude 67.5 m above and below the center.

Passengers board 2 m above ground level, so the center of the wheel must be located<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>67.5</mn><mo>+</mo><mn>2</mn><mo>=</mo><mn>69.5</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>m above ground level. The midline of the oscillation will be at 69.5 m.

The wheel takes 30 minutes to complete 1 revolution, so the height will oscillate with a period of 30 minutes.

Lastly, because the rider boards at the lowest point, the height will start at the smallest value and increase, following the shape of a vertically reflected cosine curve.

  • Amplitude:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>67</mtext><mtext>.5,</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mo>=</mo><mn>67.5</mn></mrow></annotation-xml></semantics></math>
  • Midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>69</mtext><mtext>.5,</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mo>=</mo><mn>69.5</mn></mrow></annotation-xml></semantics></math>
  • Period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mtext>30,</mtext><mtext> </mtext></mrow></annotation-xml></semantics></math>so<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>B</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π 30 = π 15
  • Shape:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>−cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )

An equation for the rider’s height would be

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mn>67.5</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 15 t )+69.5

where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is in minutes and<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is measured in meters.

Access these online resources for additional instruction and practice with graphs of sine and cosine functions.

Key Equations

 
Sinusoidal functions <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mtable columnalign="left"><mtr><mtd><mi>f</mi><mrow><mo>(</mo></mrow></mtd></mtr></mtable></annotation-xml></semantics></math> x )=Asin( Bx−C )+D f( x )=Acos( Bx−C )+D

Key Concepts

  • Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>.</mo></mrow></annotation-xml></semantics></math>
  • The function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is odd, so its graph is symmetric about the origin. The function <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>cos</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is even, so its graph is symmetric about the y-axis.
  • The graph of a sinusoidal function has the same general shape as a sine or cosine function.
  • In the general formula for a sinusoidal function, the period is<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π | B | . See [link].
  • In the general formula for a sinusoidal function,<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A | represents amplitude. If<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |>1, the function is stretched, whereas if<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>|</mo></mrow></mrow></annotation-xml></semantics></math> A |<1, the function is compressed. See [link].
  • The value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B  in the general formula for a sinusoidal function indicates the phase shift. See [link].
  • The value<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>in the general formula for a sinusoidal function indicates the vertical shift from the midline. See [link].
  • Combinations of variations of sinusoidal functions can be detected from an equation. See [link].
  • The equation for a sinusoidal function can be determined from a graph. See [link] and [link].
  • A function can be graphed by identifying its amplitude and period. See [link] and [link].
  • A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. See [link].
  • Sinusoidal functions can be used to solve real-world problems. See [link], [link], and [link].

Section Exercises

Verbal

Why are the sine and cosine functions called periodic functions?

The sine and cosine functions have the property that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x+P )=f( x ) for a certain<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>This means that the function values repeat for every<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>units on the x-axis.

How does the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> compare with the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>?</mo><mtext> </mtext></mrow></annotation-xml></semantics></math> Explain how you could horizontally translate the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math> to obtain<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

For the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo>+</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi><mo>,</mo></mrow></annotation-xml></semantics></math>what constants affect the range of the function and how do they affect the range?

The absolute value of the constant<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>(amplitude) increases the total range and the constant<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>(vertical shift) shifts the graph vertically.

How does the range of a translated sine function relate to the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>A</mi><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mi>B</mi><mi>x</mi><mo>+</mo><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi><mo>?</mo></mrow></annotation-xml></semantics></math>

How can the unit circle be used to construct the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>t</mi><mo>?</mo></mrow></annotation-xml></semantics></math>

At the point where the terminal side of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>intersects the unit circle, you can determine that the<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>t</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>equals the y-coordinate of the point.

Graphical

For the following exercises, graph two full periods of each function and state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>></mo><mn>0.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>Round answers to two decimal places if necessary.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>sin</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3 cos x

A graph of (2/3)cos(x). Graph has amplitude of 2/3, period of 2pi, and range of [-2/3, 2/3].

amplitude:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 2 3 ; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>maximum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3  occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>minimum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 3  occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>π</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>for one period, the graph starts at 0 and ends at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>3</mn><mi>sin</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mi>sin</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

A graph of 4sin(x). Graph has amplitude of 4, period of 2pi, and range of [-4, 4].

amplitude: 4; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>maximum<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>4</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2 ; minimum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3π 2 ; one full period occurs from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>2</mn><mi>π</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>cos</mi><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=cos( 2x )

A graph of cos(2x). Graph has amplitude of 1, period of pi, and range of [-1,1].

amplitude: 1; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>π</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>maximum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>π</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>minimum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mo>−</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 2 ; one full period is graphed from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mi>π</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mtext> </mtext><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 1 2 x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>4</mn><mtext> </mtext><mi>cos</mi><mo stretchy="false">(</mo><mi>π</mi><mi>x</mi><mo stretchy="false">)</mo></mrow></annotation-xml></semantics></math>

A graph of 4cos(pi*x). Grpah has amplitude of 4, period of 2, and range of [-4, 4].

amplitude: 4; period: 2; midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>maximum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>4</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>minimum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>1</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>3</mn><mtext> </mtext><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 6 5 x )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>3</mn><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mn>8</mn><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>4</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mn>5</mn></mrow></annotation-xml></semantics></math>

A graph of 3sin(8(x+4))+5. Graph has amplitude of 3, range of [2, 8], and period of pi/4.

amplitude: 3; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> π 4 ; midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>5</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>maximum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>8</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0.12</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>minimum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0.516</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>horizontal shift:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mn>4</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>vertical translation 5; one period occurs from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> π 4

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>2</mn><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mn>3</mn><mi>x</mi><mo>−</mo><mn>21</mn><mo stretchy="false">)</mo><mo>+</mo><mn>4</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>y</mi><mo>=</mo><mn>5</mn><mtext> </mtext><mi>sin</mi><mo stretchy="false">(</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>20</mn><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn></mrow></annotation-xml></semantics></math>

A graph of 5sin(5x+20)-2. Graph has an amplitude of 5, period of 2pi/5, and range of [-7,3].

amplitude: 5; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> 2π 5 ; midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>−2</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>maximum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>3</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0.08</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>minimum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>−7</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0.71;</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>phase shift:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>−4</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>vertical translation:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>−2;</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>one full period can be graphed on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2π 5

For the following exercises, graph one full period of each function, starting at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>For each function, state the amplitude, period, and midline. State the maximum and minimum y-values and their corresponding x-values on one period for<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>></mo><mn>0.</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary.

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )=2sin( t− 5π 6 )

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t+ π 3 )+1

A graph of -cos(t+pi/3)+1. Graph has amplitude of 1, period of 2pi, and range of [0,2]. Phase shifted pi/3 to the left.

amplitude: 1 ; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>2</mn><mi>π</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>1</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>maximum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>=</mo><mn>2.09</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>maximum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>2</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mn>2.09</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>minimum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mn>5.24</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>phase shift:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> π 3 ; vertical translation: 1; one full period is from<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mn>0</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>to<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mn>2</mn><mi>π</mi></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )=4cos( 2( t+ π 4 ) )−3

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t )=−sin( 1 2 t+ 5π 3 )

A graph of -sin((1/2)*t + 5pi/3). Graph has amplitude of 1, range of [-1,1], period of 4pi, and a phase shift of -10pi/3.

amplitude: 1; period:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mn>4</mn><mi>π</mi><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>maximum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mn>11.52</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>minimum:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mo>−</mo><mn>1</mn><mtext> </mtext></mrow></annotation-xml></semantics></math>occurs at<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>t</mi><mo>=</mo><mn>5.24</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>phase shift:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo>−</mo><mfrac/></mrow></annotation-xml></semantics></math> 10π 3 ; vertical shift: 0

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=4sin( π 2 ( x−3 ) )+7

Determine the amplitude, midline, period, and an equation involving the sine function for the graph shown in [link].

<figure class="small" id="Figure_06_01_218">A sinusoidal graph with amplitude of 2, range of [-5, -1], period of 4, and midline at y=-3.</figure>

amplitude: 2; midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mo>−</mo><mn>3</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>period: 4; equation:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mn>2</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 x )−3

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link].

<figure class="small" id="Figure_06_01_219">A graph with a cosine parent function, with amplitude of 3, period of pi, midline at y=-1, and range of [-4,2]</figure>

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link].

<figure class="small" id="Figure_06_01_220">A graph with a cosine parent function with an amplitude of 2, period of 5, midline at y=3, and a range of [1,5].</figure>

amplitude: 2; period: 5; midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>3</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>equation:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>2</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> 2π 5 x )+3

Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link].

<figure class="small" id="Figure_06_01_221">A sinusoidal graph with amplitude of 4, period of 10, midline at y=0, and range [-4,4].</figure>

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link].

<figure class="small" id="Figure_06_01_222">A graph with cosine parent function, range of function is [-4,4], amplitude of 4, period of 2.</figure>

amplitude: 4; period: 2; midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>0</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>equation:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>4</mn><mi>cos</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π( x− π 2 ) )

Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link].

<figure class="small" id="Figure_06_01_223">A graph with sine parent function. Amplitude 2, period 2, midline y=0</figure>

Determine the amplitude, period, midline, and an equation involving cosine for the graph shown in [link].

<figure class="small" id="Figure_06_01_224">A graph with cosine parent function. Amplitude 2, period 2, midline y=1</figure>

amplitude: 2; period: 2; midline<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>1</mn><mo>;</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>equation:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=2cos( πx )+1

Determine the amplitude, period, midline, and an equation involving sine for the graph shown in [link].

<figure class="small" id="Figure_06_01_225">A graph with a sine parent function. Amplitude 1, period 4 and midline y=0.</figure>

Algebraic

For the following exercises, let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ),solve<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=0.

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ),solve<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )= 1 2 .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 6 , 5π 6

Evaluate<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 2 ).

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo><mo>,</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 2 2 . Find all values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 4 , 3π 4

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ),the maximum value(s) of the function occur(s) at what x-value(s)?

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ),the minimum value(s) of the function occur(s) at what x-value(s)?

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mrow><mn>3</mn><mi>π</mi></mrow></mfrac></mrow></annotation-xml></semantics></math> 2

Show that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>−</mi><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>.</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>This means that<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>is an odd function and possesses symmetry with respect to ________________.

For the following exercises, let<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ),solve the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>=</mo><mn>0.</mn></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 2 , 3π 2

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ),solve<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 1 2 .

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ),find the x-intercepts of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 2 , 3π 2

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ),find the x-values at which the function has a maximum or minimum value.

On<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ),solve the equation<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> 3 2 .

<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mfrac><mi>π</mi></mfrac></mrow></annotation-xml></semantics></math> 6 , 11π 6

Technology

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ]. Explain why the graph appears as it does.

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>+</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −100,100 ]. Did the graph appear as predicted in the previous exercise?

The graph appears linear. The linear functions dominate the shape of the graph for large values of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

A sinusoidal graph that increases like the function y=x, shown from 0 to 100.

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>on<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> 0,2π ] and verbalize how the graph varies from the graph of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mtext> </mtext><mi>x</mi><mo>.</mo></mrow></annotation-xml></semantics></math>

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mtext> </mtext><mi>sin</mi><mtext> </mtext><mi>x</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>on the window<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −10,10 ] and explain what the graph shows.

The graph is symmetric with respect to the y-axis and there is no amplitude because the function is not periodic.

A sinusoidal graph that has increasing peaks and decreasing lows as the absolute value of x increases.

Graph<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac/></mrow></annotation-xml></semantics></math> sin x x  on the window<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mrow><mo>[</mo></mrow></mrow></annotation-xml></semantics></math> −5π,5π ] and explain what the graph shows.

Real-World Applications

A Ferris wheel is 25 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. The function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t ) gives a person’s height in meters above the ground t minutes after the wheel begins to turn.

  1. Find the amplitude, midline, and period of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t ).
  2. Find a formula for the height function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>h</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> t ).
  3. How high off the ground is a person after 5 minutes?
  1. Amplitude: 12.5; period: 10; midline:<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mn>13.5</mn><mo>;</mo></mrow></annotation-xml></semantics></math>
  2. <math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mn>12.5</mn><mi>sin</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> π 5 ( t−2.5 ) )+13.5;
  3. 26 ft

Glossary

amplitude
the vertical height of a function; the constant<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>A</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>appearing in the definition of a sinusoidal function
midline
the horizontal line<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>y</mi><mo>=</mo><mi>D</mi><mo>,</mo><mtext> </mtext></mrow></annotation-xml></semantics></math>where<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>D</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>appears in the general form of a sinusoidal function
periodic function
a function<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x ) that satisfies<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x+P )=f( x ) for a specific constant<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>P</mi><mtext> </mtext></mrow></annotation-xml></semantics></math>and any value of<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>x</mi></mrow></annotation-xml></semantics></math>
phase shift
the horizontal displacement of the basic sine or cosine function; the constant<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mfrac/></mrow></annotation-xml></semantics></math> C B
sinusoidal function
any function that can be expressed in the form<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=Asin( Bx−C )+D or<math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation-xml encoding="MathML-Content"><mrow><mtext> </mtext><mi>f</mi><mrow><mo>(</mo></mrow></mrow></annotation-xml></semantics></math> x )=Acos( Bx−C )+D