Transcript C) + D

CHAPTER 6:
The Trigonometric Functions
6.1
6.2
6.3
6.4
6.5
6.6
The Trigonometric Functions of Acute Angles
Applications of Right Triangles
Trigonometric Functions of Any Angle
Radians, Arc Length, and Angular Speed
Circular Functions: Graphs and Properties
Graphs of Transformed Sine and Cosine
Functions
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6.6
Graphs of Transformed Sine and Cosine
Functions



Graph transformations of y = sin x and y = cos x in the
form y = A sin (Bx – C) + D and
y = A cos (Bx – C) + D and determine the amplitude,
the period, and the phase shift.
Graph sums of functions.
Graph functions (damped oscillations) found by
multiplying trigonometric functions by other
functions.
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Variations of the Basic Graphs
We are interested in the graphs of functions in
the form
y = A sin (Bx – C) + D
and
y = A cos (Bx – C) + D
where A, B, C, and D are all constants. These
constants have the effect of translating,
reflecting, stretching, and shrinking the basic
graphs.
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The Constant D
Let’s observe the effect of the constant D.
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The Constant D
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The Constant D
The constant D in
y = A sin (Bx – C) + D
and
y = A cos (Bx – C) + D
translates the graphs up D units if D > 0 or down
|D| units if D < 0.
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The Constant A
Let’s observe the effect of the constant A.
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The Constant A
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The Constant A
If |A| > 1, then there will be a vertical stretching.
If |A| < 1, then there will be a vertical shrinking.
If A < 0, the graph is also reflected across the xaxis.
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Amplitude
The amplitude of the graphs of
y = A sin (Bx – C) + D
and
y = A cos (Bx – C) + D
is |A|.
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The Constant B
Let’s observe the effect of the constant B.
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The Constant B
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The Constant B
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The Constant B
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The Constant B
If |B| < 1, then there will be a horizontal
stretching.
If |B| > 1, then there will be a horizontal
shrinking.
If B < 0, the graph is also reflected across the
y-axis.
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Period
The period of the graphs of
y = A sin (Bx – C) + D
and
y = A cos (Bx – C) + D
2
.
is
B
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Period
The period of the graphs of
y = A csc (Bx – C) + D
and
y = A sec (Bx – C) + D
2
.
is
B
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Period
The period of the graphs of
y = A tan (Bx – C) + D
and
y = A cot (Bx – C) + D
is

B
.
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The Constant C
Let’s observe the effect of the constant C.
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The Constant C
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The Constant C
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The Constant C
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The Constant C
If B = 1, then
if |C| < 0, then there will be a horizontal
translation of |C| units to the right, and
if |C| > 0, then there will be a horizontal
translation of |C| units to the left.
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Combined Transformations
It is helpful to rewrite
y = A sin (Bx – C) + D
and
y = A cos (Bx – C) + D
as
and
C
 
y  Asin  B  x     D
B 
 
C
 
y  A cos  B  x     D
B 
 
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Phase Shift
The phase shift of the graphs
C
 
y  Asin Bx  C   D  Asin  B  x     D
B 
 
and
C
 
y  A cos Bx  C   D  A cos  B  x     D
B 
 
C
is the quantity .
B
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Phase Shift
If C/B > 0, the graph is translated to the right
|C/B| units.
If C/B < 0, the graph is translated to the right
|C/B| units.
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Transformations of Sine and Cosine
Functions
To graph
C
 
y  Asin Bx  C   D  Asin  B  x     D
B 
 
and
C
 
y  A cos Bx  C   D  A cos  B  x     D
B 
 
follow the steps listed below in the order in
which they are listed.
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Transformations of Sine and Cosine
Functions
1. Stretch or shrink the graph horizontally
according to B.
|B| < 1
|B| > 1
B<0
Stretch horizontally
Shrink horizontally
Reflect across the y-axis
2
.
The period is
B
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Transformations of Sine and Cosine
Functions
2. Stretch or shrink the graph vertically
according to A.
|A| < 1
|A| > 1
A<0
Shrink vertically
Stretch vertically
Reflect across the x-axis
The amplitude is A.
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Transformations of Sine and Cosine
Functions
3. Translate the graph horizontally according
to C/B.
C
0
B
C
units to the left
B
C
0
B
C
units to the right
B
C
The phase shift is .
B
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Transformations of Sine and Cosine
Functions
4. Translate the graph vertically according
to D.
D<0
|D| units down
D>0
D units up
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Slide 6.6 - 32
Example
Sketch the graph of y  3sin 2x   / 2   1.
Find the amplitude, the period, and the phase shift.
Solution:
 


   
y  3sin  2x    1  3sin  2  x        1

 4 
2
 
Amplitude  A  3  3
2
2
Period 


B
2
C  2

Phase shift  

B
2
4
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Slide 6.6 - 33
Example
Solution continued
To create the final graph, we begin with the basic sine
curve, y = sin x.
Then we sketch graphs of each of the following
equations in sequence.
1. y  sin 2x
2. y  3sin 2x
 
   
3. y  3sin  2  x      
 4 
 
 
   
4. y  3sin  2  x        1
 4 
 
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Example
Solution continued
y  sin x
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Example
Solution continued
1. y  sin 2x
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Example
Solution continued
2. y  3sin 2x
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Slide 6.6 - 37
Example
Solution continued
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 
   
3. y  3sin  2  x      
 4 
 
Slide 6.6 - 38
Example
Solution continued
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 
   
4. y  3sin  2  x        1
 4 
 
Slide 6.6 - 39
Example
Graph: y = 2 sin x + sin 2x
Solution:
Graph: y = 2 sin x and y = sin 2x on the same axes.
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Slide 6.6 - 40
Example
Solution continued
Graphically add some y-coordinates, or ordinates, to
obtain points on the graph that we seek.
At x = π/4, transfer h up to add it to 2 sin x, yielding P1.
At x = – π/4, transfer m down to add it to 2 sin x,
yielding P2.
At x = – 5π/4, add the negative ordinate of sin 2x to the
positive ordinate of 2 sin x, yielding P3.
This method is called addition of ordinates, because
we add the y-values (ordinates) of y = sin 2x to the yvalues (ordinates) of y = 2 sin x.
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Slide 6.6 - 41
Example
Solution continued
The period of the sum 2 sin x + sin 2x is 2π, the least
common multiple of 2π and π.
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Example
Sketch a graph of f x   e x 2 sin x.
Solution
f is the product of two functions g and h, where
g x   e x 2
and
h x   sin x
To find the function values, we can multiply ordinates.
Start with
1  sin x  1
ex 2  ex 2 sin x  ex 2
The graph crosses the x-axis at values of x for which
sin x = 0, kπ for integer values of k.
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Example
Solution continued
f is constrained between the graphs of y = –e–x/2 and
y = e–x/2. Start by graphing these functions using
dashed lines.
Since f(x) = 0 when x = kπ, k an integer, we mark
those points on the graph.
Use a calculator to compute other function values.
The graph is on the next slide.
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Slide 6.6 - 44
Example
Solution continued
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