8-4 Graphs of Sine and Cosine (Presentation)

8/7/2019 8-4 Graphs of Sine and Cosine (Presentation)

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8-4 Graphs of Sine and Cosine

Unit 8 Trigonometric and Circular Functions


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Concepts and Objectives

Graphs of the Sine and Cosine Functions (Obj. #29)

Be able to graph the sine and cosine functions on theinterval [2, 2]

Be able to identify how the graphs of the sine and

Amplitude

Period

Vertical translation

Phase shift


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Graphing the Sine Function

Because the sine is a circularfunction, while can be

any number (positive or negative), the value of sin only goes from 1 to 1.

To graph a circular function, we can take the angle as

-

from the unit circle as our y-coordinate.


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Graphing the Sine Function

If we graph these values on a coordinate plane, we can

see how this function curves:

The key to graphing the sine function is that it includesthe point(0, 0).


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Graphing the Cosine Function

The cosine function is very similar to the sine. Lets look

at a graph of it:

The key to graphing the cosine function is that itincludes the point(0, 1).


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Changing the Basic Graph

Amplitude describes how the basic graph is changed

vertically (taller or shorter). The amplitude of a periodicfunction is half the difference between the maximum

and minimum values.

does it take for the graph to start repeating.

The period of the sine or cosine is 2.


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Amplitude

Compare the graphs of and= siny x = 2siny x

Notice that both graphs cross the x-axis at the same

places and that both graphs peak at the same places.


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Amplitude

The graph ofy= a sin xor y= a cos x, with a 0, will

have the same shape as the graph ofy= sin xor y= cos x,respectively, except with the range [|a|, |a|]. The

amplitude is |a|.

Example: What is the amplitude of ? What is

the range of the graph?= 1cos

3y x

1Amplitude: 3

1 1

Range: ,3 3


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Period

Compare the graphs ofy= cos xand y= cos 2x.

The two graphs have the same height, but the second

graph is compressed.


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Period

For b > 0, the graph ofy= sin bxwill resemble that of

y= sin x, but with period . Also, the graph of

y= cos bxwill resemble that ofy= cos x, but with period

2b

To calculate the period of a function, substitute the

coefficient ofxin for b and reduce.

b


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Sketching a Circular Graph

Example: Sketch the graph of over one

period.

The coefficient ofxis , so the period is

= 2sin2

y

1

2

=

24

1

Now, divide the interval [0, 4] into four equal parts and

evaluate the function at those values.

2


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period.

= 2sin2

y

x 0 2 3 4

2x

sin2

x

2sin2x

0

0

0

2

1

2

0

0

32

1

2

2

0

0


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period.

= 2sin2

y

Notice that when a is negative, the graph is reflected

about the x-axis.


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Translating Sine and Cosine

We have seen what the graph ofy= a sin bxlooks like.

Next, we can shift the graph vertically and/orhorizontally.

The full form of the sine function is

c affects the vertical position of the graph. A positive

c shifts the graph c units up, and a negative c shifts

the graph c units down.

dshifts the graph horizontally. (x+ d) shifts the graphdunits to the left, and (x d) shifts the graph dunits

to the right.


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Translating Sine and Cosine

With circular functions, a horizontal translation is called

a phase shift. The phase shift is the absolute value ofd.

To sketch the translated graph, you can either divide the

chart the values as before, or you can sketch the

stretched/compressed parent graph and translate it

according to c and d.

The second method is probably the easiest to do onceyou are comfortable with the basic graphs.


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Graphing Sine and Cosine Example: Graph over one period.

= +

3cos

4y x

3, 1, 0, to the left4

a b c d

= = = =


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Graphing Sine and Cosine Example: Graph over two periods.

To find the value ofb, we will have to factor out the 4 in

front of the x:

( )= + + 1 2sin 4y x

= + + 1 2sin4 4y x

2, 4, 1, to the left4

a b c d

= = = =

2 2Period:4 2b

= =


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Graphing Sine and Cosine Example: Graph over two periods.( )= + + 1 2sin 4y x

= + +

1 2sin44

y x

= = = = , , ,

4Period:

2


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Homework College Algebra (brown book)

Page 593: 15, 16, 19, 23, 34 Turn in: 16, 34

Page 606: 25, 30, 33, 36, 37, 40

Turn in: 30, 36, 40

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8-4 Graphs of Sine and Cosine (Presentation)