obj. 20 translations of sine and cosine graphs (presentation)
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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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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 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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Graphing Sine and Cosine
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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Writing Equations From Graphs
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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Writing Equations From Graphs
Example: Write an equation for the graph below.
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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Writing Equations From Graphs
Example: Write an equation for the graph below.
=
2period
b
=
2
b
= 2b
= +1 sin2y x
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Writing Equations From Graphs
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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Writing Equations From Graphs
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
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