Download - 5.5 Graphs of Other Trigonometric Functions
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5.5 Graphs of Other Trigonometric Functions
Objectives:•Understand the graph of y = sin x.•Graph variations of y = sin x.•Understand the graph of y = cos x.•Graph variations of y = cos x.•Use vertical shifts of sine and cosine curves.•Model periodic behavior.Dr .Hayk Melikyan
Department of Mathematics and CS
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The Graph of y = sinx
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Some Properties of the Graph of y = sinx
The domain is ( , ). The range is [–1, 1].
The period is 2 . The function is an odd function:sin( ) sin .x x
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Graphing Variations of y = sinx
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Example: Graphing a Variation of y = sinx
Determine the amplitude of Then graph
and for
Step 1 Identify the amplitude and the period.
The equation is of the form y = Asinx with A = 3. Thus, the amplitude is This means that the maximum value of y is 3 and the minimum value of y is – 3. The period is
3sin .y x siny x3siny x 2 .x
3siny x3.A
2 .
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Example: Graphing a Variation of y = sinx (continued)
Determine the amplitude of Then graph
and for
Step 2 Find the values of x for the five key points. To generate x-values for each of the five key points, we begin by dividing the period, by 4. The cycle begins at x1 = 0. We add quarter periods to generate x-values for each of the key points.
3sin .y x siny x3siny x 2 .x
2 ,
24 2 1 0x 2 0
2 2x
3 2 2x
4
32 2
x 5
32
2 2x
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Example: Graphing a Variation of y = sinx (continued)
Step 3 Find the values of y for the five key points.
3siny x ( , )x y
3sin 0 3 0 0y (0,0)0
2
32
2
3sin 3 1 32
y
x
3sin 3 0 0y
33sin 3 ( 1) 3
2y
3sin 2 3 0 0y
,32
,0
3, 3
2
2 ,0
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Example: Graphing a Variation of y = sinx (continued)
Determine the amplitude of Then graph
and for
Step 4 Connect the five keypoints with a smooth curve and graph one complete cycle of the given function.
3sin .y x siny x3siny x 2 .x
3siny x
siny x
amplitude = 3
period 2
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Amplitudes and Periods
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Example: Graphing a Function of the Form y = AsinBx
Determine the amplitude and period of Then graph the function for
Step 1 Identify the amplitude and the period.
The equation is of the form y = A sinBx with A = 2 and
The maximum value of y is 2, the minimum value of y is –2. The period of tells us that the graph completes one cycle from 0 to
12sin .
2y x
0 8 .x
1.
2B
44 .
2A 2 2
412
B amplitude: period:
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Example: Graphing a Function of the Form y = AsinBx (continued)
Step 2 Find the values of x for the five key points.
To generate x-values for each of the five key points, we begin by dividing the period, by 4. The cycle begins at x1 = 0. We add quarter periods to generate x-values for each of the key points.
4 ,
44 1 0x 2 0x 3 2x
4 2 3x 5 3 4x
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Example: Graphing a Function of the Form y = AsinBx (continued)
Step 3 Find the values of y for the five key points.1
2sin2
y xx ( , )x y
0
2
3
4 12sin 4 2sin 2 2 0 0
2y
1 32sin 3 2sin 2 ( 1) 2
2 2y
12sin 2 2sin 2 0 0
2y
12sin 2sin 2 1 2
2 2y
12sin 0 2sin 0 0
2y
(0,0)
,2
2 ,0
3 , 2
4 ,0
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Example: Graphing a Function of the Form y = AsinBx (continued)
Determine the amplitude and period of Then graph the function for
Step 4 Connect the five key pointswith a smooth curve and graphone complete cycle of the given function.
12sin .
2y x
0 8 .x
amplitude = 2
period 4
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Example: Graphing a Function of the Form y = AsinBx (continued)
Determine the amplitude and period of Then graph the function for
Step 5 Extend the graph in step 4 to the left or right as desired. We will extend the graph to include
the interval
12sin .
2y x
0 8 .x
0 8 .x
amplitude = 2
period 4
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The Graph of y = Asin(Bx – C)
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Example: Graphing a Function of the Form y = Asin(Bx – C)
Determine the amplitude, period, and phase shift of
Then graph one period of the function.
Step 1 Identify the amplitude, the period, and the
phase shift.
3sin 2 .3
y x
3sin 23
y x
3, 2,
3A B C
amplitude: period:
phase shift:
3 3A 2 22B
132 3 2 6
CB
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Example: Graphing a Function of the Form y = Asin(Bx – C) (continued)
Step 2 Find the values of x for the five key points.period
4 4 1 6
x 2
2 3 56 4 12 12 12
x
3
5 5 3 8 212 4 12 12 12 3
x
4
2 8 3 113 4 12 12 12
x
5
11 11 3 14 712 4 12 12 12 6
x
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Example: Graphing a Function of the Form y = Asin(Bx – C) (continued)
Step 3 Find the values of y for the five key points.
x 3sin 23
y x
( , )x y
6
512
3sin 2 3sin 3sin 0 3 0 06 3 3 3
y
5 5 2 33sin 2 3sin 3sin
12 3 6 6 6
3sin 3 1 32
y
23
2 4 33sin 2 3sin 3sin
3 3 3 3 3
3sin 3 0 0
y
,06
5,3
12
2,0
3
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Example: Graphing a Function of the Form y = Asin(Bx – C) (continued)
Step 3 (cont) Find the values of y for the five key points.
x 3sin 23
y x
( , )x y
1112
76
11 11 93sin 2 3sin 3sin
12 3 6 6 6
33sin 3 ( 1) 3
2
y
7 7 63sin 2 3sin 3sin
6 3 3 3 3
3sin 2 3 0 0
y
11, 3
12
7,0
6
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Step 4 Connect the five key points with a smooth curve and graph one complete cycle of the function
Example: Graphing a Function of the Form y = Asin(Bx – C) (continued)
amplitude = 3
period
phase shift6
3sin 2 .3
y x
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The graph of y = cosx
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Some Properties of the Graph of y = cosx
The domain is ( , ). The range is [–1, 1].
The period is 2 . The function is an even function:cos( ) cos .x x
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Sinusoidal Graphs
The graph of is the graph ofwith a phase shiftof
The graphs of sinefunctions and cosinefunctions are calledsinusoidal graphs.
siny xcosy x
.2
cos sin2
x x
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Graphing Variations of y = cosx
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Example: Graphing a Function of the Form y = AcosBx
Determine the amplitude and period of
Then graph the function for
Step 1 Identify the amplitude and the period.
4cos .y x2 2.x
amplitude: period:
cos 4cosy A Bx x
4 4A 2 22
B
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Example: Graphing a Function of the Form y = AcosBx (continued)
Determine the amplitude and period of
Then graph the function for
Step 2 Find the values of x for the five key points.
4cos .y x2 2.x
period 2 14 4 2
1 0x 2
1 10
2 2x
3
1 11
2 2x 4
1 31
2 2x 5
3 12
2 2x
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Example: Graphing a Function of the Form y = AcosBx (continued)
Step 3 Find the values of y for the five key points.x ( , )x y4cosy x
0
12
1
4cos( 0) 4cos0 4 1 4y
14cos 4cos 4 0 0
2 2y
4cos 1 4cos 4 ( 1) 4y
(0, 4)
1,0
2
(1,4)
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Example: Graphing a Function of the Form y = AcosBx (continued)
Step 3 Find the values of y for the five key points.x ( , )x y4cosy x
32
2
3 34cos 4cos 4 0 0
2 2y
4cos 2 4cos(2 ) 4 1 4y
3,0
2
2, 4
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Example: Graphing a Function of the Form y = AcosBx (continued)
Determine the amplitude and period of
Then graph the function for
Step 4 Connect the five key
points with a smooth curve
and graph one complete
cycle of the given function.
4cos .y x2 2.x
1 2 3
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
amplitude = 4
period = 2
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Example: Graphing a Function of the Form y = AcosBx (continued)
Determine the amplitude and period of
Then graph the function for
Step 5 Extend the graph
to the left or right
as desired.
4cos .y x2 2.x
-2 -1 1 2
-5
-4
-3
-2
-1
1
2
3
4
5
x
y
amplitude = 4
period = 2
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The Graph of y = Acos(Bx – C)
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Example: Graphing a Function of the Form y = Acos(Bx – C)
Determine the amplitude, period, and phase shift of
Then graph one period of the function.
Step 1 Identify the amplitude, the period, and the
phase shift.
3cos 2 .
2y x
3cos( ) cos(2 )
2y A Bx C x
amplitude: period:
phase shift:
3 32 2
2 22B
2 2CB
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Example: Graphing a Function of the Form y = Acos(Bx – C) (continued)
Determine the amplitude, period, and phase shift of
Then graph one period of the function.
Step 2 Find the x-values for the five key points.
3cos 2 .
2y x
period4 4
1 2x
2 2 4 4x
3 04 4
x 4 0
4 4x
5
24 4 4 2
x
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Example: Graphing a Function of the Form y = Acos(Bx – C) (continued)
Step 3 Find the values of y for the five key points.
x 3cos 2
2y x ( , )x y
2
4
3 3 3cos 2 cos( ) cos0
2 2 2 2
3 31
2 2
y
3 3 3cos 2 cos cos
2 4 2 2 2 2
30 0
2
y
3,
2 2
,04
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Example: Graphing a Function of the Form y = Acos(Bx – C) (continued)
Step 3 (cont) Find the values of y for the five key points.
x 3cos 2
2y x ( , )x y
0
4
3 3 3 3cos 2 0 cos( ) ( 1)
2 2 2 2y
3 3 3 3cos 2 cos cos
2 4 2 2 2 2
30 0
2
y
30,
2
,04
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Example: Graphing a Function of the Form y = Acos(Bx – C) (continued)
Step 3 (cont) Find the values of y for the five key points.
x 3cos 2
2y x ( , )x y
2
3 3 3cos 2 cos( ) cos(2 )
2 2 2 2
3 31
2 2
y
3,
2 2
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Example: Graphing a Function of the Form y = Acos(Bx – C) (continued)
Determine the amplitude, period, and phase shift of
Then graph one period of the function.
Step 4 Connect the five
key points with
a smooth curve
and graph one
complete cycle of the
given function.
3cos 2 .
2y x
amplitude32
period
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Vertical Shifts of Sinusoidal Graphs
For sinusoidal graphs of the form
and
the constant D causes a vertical shift in the graph.
These vertical shifts result in sinusoidal graphs oscillating about the horizontal line y = D rather than about the x-axis.
The maximum value of y is
The minimum value of y is
sin( )y A Bx C D cos( ) y A Bx C D
.D A.D A
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Example: A Vertical Shift
Graph one period of the function
Step 1 Identify the amplitude, period, phase shift, and vertical shift.
2cos 1.y x
cos( )y A Bx C D 2cos 1y x
amplitude: period:
phase shift:
2 2A 2 22
1B
0 vertical shift: one unit upward
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Example: A Vertical Shift
Graph one period of the function
Step 2 Find the values of x for the five key points.
2cos 1.y x
period 24 4 2
1 0x 2 02 2
x
3 2 2x
4
32 2
x 5
32
2 2x
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Example: A Vertical Shift
Step 3 Find the values of y for the five key points.
x ( , )x y2cos 1y x
0
2
2cos0 1 2 1 1 3y
2cos 1 2 0 1 12
y
2cos 1 2 ( 1) 1 1y
32 3
2cos 1 2 0 1 12
y
0,3
,12
, 1
3,1
2
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Example: A Vertical Shift
Step 3 (cont) Find the values of y for the five key points.
x ( , )x y2cos 1y x
2 2cos 2 1 2 1 1 3y 2 ,3
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Example: A Vertical Shift
Graph one period of the function
Step 4 Connect the five
key points with
a smooth curve
and graph one
complete cycle of
the given function.
2cos 1.y x
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Example: Modeling Periodic Behavior
A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y = Asin(Bx – C) + D to model the hours of daylight.
Because the hours of daylight range from a minimum of 10 to a maximum of 14, the curve oscillates about the middle value, 12 hours. Thus, D = 12.
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Example: Modeling Periodic Behavior (continued)
A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y = Asin(Bx – C) + D to model the hours of daylight.
The maximum number of hours of daylight is 14, which is 2 hours more than 12 hours. Thus, A, the amplitude, is 2; A = 2.
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Example: Modeling Periodic Behavior
A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y = Asin(Bx – C) + D to model the hours of daylight.
One complete cycle occurs over a period of 12 months.2
12B 12 2B
212 6
B
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Example: Modeling Periodic Behavior
The starting point of the cycle is March, x = 3.
The phase shift is
3CB
3
6
C 3
36 6 2
C
.2
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Example: Modeling Periodic Behavior
A region that is 30° north of the Equator averages a minimum of 10 hours of daylight in December. Hours of daylight are at a maximum of 14 hours in June. Let x represent the month of the year, with 1 for January, 2 for February, 3 for March, and 12 for December. If y represents the number of hours of daylight in month x, use a sine function of the form y = Asin(Bx – C) + D to model the hours of daylight.
2, , , 126 2
A B C D
The equation that models the hours of daylight is
2sin 12.6 2
y x