obj. 20 translations of sine and cosine graphs (presentation)

8/3/2019 Obj. 20 Translations of Sine and Cosine Graphs (Presentation)

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Obj. 20 Translating Graphs of

Sine and Cosine

Unit 5 Trigonometric and Circular Functions


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Concepts and Objectives

Graphs of the Sine and Cosine Functions (Obj. #20)

Be able to identify how the graphs of the sine andcosine change due to changes in

Amplitude

Period Vertical translation

Phase shift


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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.

( )= + siny c a b x d


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Translating Sine and Cosine

With circular functions, a horizontal translation is called

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

To sketch the translated graph, you can either divide the

interval into four parts (eight parts for two periods) and

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 left

4a b c d


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Example: Graph over two periods.

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

front of thex:

( )= + + 1 2sin 4y x

= + +1 2sin4 4y x

= = = =2, 4, 1, to the left

4a b c d

= =2 2Period:

4 2b


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

= + +1 2sin4

4y x

= = = = 2, 4, 1,

4a b c d

Period:

2


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Writing Equations From Graphs

Now that we know how the different factors affect the

graphs of sine and cosine, we can write the equationsfrom the graphs.

Remember, from

a is the amplitude (height)

b is the period (width)

c is the vertical shift (up or down)

dis the phase shift (left or right)

Also, recall that sine goes through the origin and cosine

doesnt.

( )= + siny c a b x d


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Example: Write an equation for the graph below.

1. Find the middle of the

graph. This tells us that

c = 1 and a = 1.

2. Shift the graph so that

the middle is on thex-

axis.

3. Since the graph goesthrough the origin, we

will use sine.


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4. Since the graph goes

through the origin, we

dont have to worry

about a phase shift, so

d= 0.

5. One period of the graph

is from 0 to , so we canuse that to calculate b.


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=

2period

b

=

2

b

= 2b

= +1 sin2y x


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Example: Write an equation using cosine for the graph.

1. c = 2,

2. This time we have aphase shift. Since we

have to use cosine, we

will shift the graph over

/4 to the right.

=1

2a

=

4d


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Example: Write an equation using cosine for the graph.

3. Since cosine normally

starts above thex-axis,

this graph has a

negative a.

4. The period goes from 0

to 2, so b is 1.

= +

12 cos

2 4y x


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Homework

College Algebra

Page 605: 19-22, 24-30 (even), 36-42 (even) HW: 20, 24, 28, 30, 36, 40

obj. 20 translations of sine and cosine graphs (presentation)

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