Graphing translations of trig functions
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Chapter 14 Trigonometric Graphs and Identities14.1 Graphing Trig Functions
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Graphs of Sine and Cosine FunctionsGraphs of Sine and Cosine Functions
Objectives:Objectives:
• to sketch the graphs of basic sine and cosine functions;to sketch the graphs of basic sine and cosine functions;• to use amplitude and period to help sketch the graphs of sine to use amplitude and period to help sketch the graphs of sine
and cosine functions;and cosine functions;• to sketch translations of the graphs of sine and cosine to sketch translations of the graphs of sine and cosine
functions;functions;• to use sine and cosine functions to model real-life data.to use sine and cosine functions to model real-life data.
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Graph of Sine
2 2 2
2
2
32
3
1
1
Domain: All Real NumbersRange: [-1, 1]
Symmetry: Origin
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Graph of Cosine
2 2 2
2
2
32
3
1
1
Domain: All Real NumbersRange: [-1, 1]
Symmetry: y-axis
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Amplitude
The amplitude of the functions y = Asin and y = Acos is the absolute value of A.
The amplitude of a graph can be described as the absolute value of one-half the difference of the maximum and
minimum function values.
2 2 2
2
2
32
3
1
1
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Example
State the amplitude of the function y = 3cos. Graph both y = 3cos and y = cos on the same coordinate plane.
A = 3
2 2 2
2
2
32
3
1
1
2
3
2
3
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Period
The period of a function is the distance required to complete one full cycle.
kP
2
The period of the functions y = sink and y = cosk is given by:
2 2 2
2
2
32
3
1
1
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Example
State the period of the function y = cos4. Graph both y = cos4 and y = cos on the same coordinate plane.
24
2 P
2 22
2
2
32
3
1
2
3
2
3
1
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Amplitudes and Periods
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14.2 Translations of Trig Graphs
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The graph of y = A sin (Bx – C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C/B. The number C/B is called the phase shift.
amplitude = | A|
period = 2 /B.
The graph of y = A sin (Bx – C) is obtained by horizontally shifting the graph of y = A sin Bx so that the starting point of the cycle is shifted from x = 0 to x = C/B. The number C/B is called the phase shift.
amplitude = | A|
period = 2 /B.
x
y
Amplitude: | A|
Period: 2/B
y = A sin Bx
Starting point: x = C/B
The Graph of y = Asin(Bx - C)
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Example
Determine the amplitude, period, and phase shift of y = 2sin(3x-)
Solution:
Amplitude = |A| = 2
period = 2/B = 2/3
phase shift = C/B = /3
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Example cont.
-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
-3
-2
-1
1
2
3
• y = 2sin(3x- )
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dcbxa )sin(Amplitude
Period:
2π/b Phase Shift:
c/b
Vertical
Shift
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Phase Shift
A phase shift is simply a horizontal change.
The phase shift of the function y = A cosk( + C) is C. If C > 0, the shift is to the left. If C < 0, the shift is to the right.
2 2 2
2
2
32
3
1
1
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Example
4cos
y
State the phase shift of the following function. Then sketch the function and y = cos on the same coordinate plane.
4
PS
2 2 2
2
2
32
3
1
1
2
3
2
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Vertical Shift
A vertical shift is simply a vertical change.
The vertical shift of the function y = Asink + C is C. If C > 0, the shift is up. If C < 0, the shift is down.
2 2 2
2
2
32
3
1
1
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Example
2cos y
State the vertical shift of the following function. Then sketch the function and y = cos on the same coordinate plane.
2VS
2 2 2
2
2
32
3
1
1
2
3
2
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Practice
State the amplitude, period, phase shift, and vertical shift of the following function.
4cos6 y
4
2
6
VS
PS
P
A
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Finding a Trigonometric Model
Throughout the day, the depth of water at the end of a dock in Bar Harbor, Maine, varies with the tides. The table below shows the depths (in feet) at various times during the morning. (a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon can it
safely dock?Time, Time, tt Depth, Depth, yy
MidnightMidnight 3.43.4
2 A.M.2 A.M. 8.78.7
4 A.M.4 A.M. 11.311.3
6 A.M.6 A.M. 9.19.1
8 A.M.8 A.M. 3.83.8
10 A.M.10 A.M. 0.10.1
NoonNoon 1.21.2
Begin by graphing the data. Let the time be the independent variable and the depth be the
dependent variable.
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Finding a Trigonometric Model(a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon
can it safely dock?
The model fits either a sine or
cosine model. Select cosine.
10
8
6
4
2
5 10
The amplitude is half the difference of the
max and min.a = 0.5(11.3 – 0.1)
a = 5.6
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Finding a Trigonometric Model(a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon
can it safely dock?
The period can be found by doubling
the difference of the low time and high
time.
10
8
6
4
2
5 10
p = 2(10 – 4)
p = 12
k
212
6
k
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Finding a Trigonometric Model(a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon
can it safely dock?
The high point should occur at the
origin. Thus there is a shift of 4 to the
right.
10
8
6
4
2
5 10
With an amplitude of 5.6, the high point has been raised by
5.7
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Finding a Trigonometric Model(a) Use a trigonometric function to model this data. (b) Find the depths at 9 A.M. and 3 P.M. (c) A boat needs at least 10 feet of water to moor at the dock. During what times in the afternoon
can it safely dock? Given all these changes, the model
of best fit is10
8
6
4
2
5 10
7.546
cos6.5 ty
7.53
2
6cos6.5
ty
fty
fty
57.1015
84.09
The depth is at least 10 feet from 2:42 P.M. to
5:18 P.M.
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HomeworkHomework14.1 pg 767 #’s 15, 18, 21, 24, 27, 30, 3314.1 pg 767 #’s 15, 18, 21, 24, 27, 30, 33
14.2 pg 774 #’s 19-24 all14.2 pg 774 #’s 19-24 all