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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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Sketching a Circular Graph
Example: Sketch the graph of over one
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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Sketching a Circular Graph
Example: Sketch the graph of over one
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