Chapter09 :Sinusoids and Phasors electric circuit textbook solution
Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be...
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Transcript of Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be...
![Page 1: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/1.jpg)
Sinusoids and Sinusoids and TransformationsTransformations
Sec. 4.4bSec. 4.4b
![Page 2: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/2.jpg)
Definition: Sinusoid
sin[ ( )]f x a b x h k
A function is a sinusoid if it can be written in the form
where a, b, c, and d are constants and neither a nor b is 0.
In general, any transformation of a sine function (or the graphof such a function – such as cosine) is a sinusoid.
sin( )f x a bx c d
This is the format that we are used to seeing, thus it is OK to continue using this format…I use this format.
![Page 3: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/3.jpg)
TransformationsThere is a special vocabulary for describing our traditionalgraphical transformations when applied to sinusoids…
Horizontal stretches and shrinks affect the periodand the frequency.
Vertical stretches and shrinks affect the amplitude.
Horizontal translations bring about phase shifts.
![Page 4: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/4.jpg)
Definition: Amplitude of a Sinusoid
sin[ ( )]f x a b x h k The amplitude of the sinusoid
is a
Similarly, the amplitude of
cos[ ( )]f x a b x h k is a
Graphically, the amplitude is half the height of the wave.
![Page 5: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/5.jpg)
TransformationsFind the amplitude of each function and use the language oftransformations to describe how the graphs are related.
(a)1 cosy x (b) 2
1cos2
y x (c) 3 3cosy x
Amplitudes: (a) 1, (b) 1/2, (c) |–3| = 3
The graph of y is a vertical shrink of the graph of y by 1/2.2 1
The graph of y is a vertical stretch of the graph of y by 3,and a reflection across the x-axis, performed in either order.
3 1
Confirm these answers graphically!!!Confirm these answers graphically!!!
![Page 6: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/6.jpg)
Definition: Period of a Sinusoid
sin[ ( )]f x a b x h k The period of the sinusoid
is 2 b
Similarly, the period of
cos[ ( )]f x a b x h k is2 b
Graphically, the period is the length of one full cycle ofthe wave.
![Page 7: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/7.jpg)
TransformationsFind the period of each function and use the language oftransformations to describe how the graphs are related.
(a)1 siny x
(b) 2 2sin3
xy
(c) 3 3sin 2y x
Periods
2
2 1 3 6
2 2
![Page 8: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/8.jpg)
TransformationsFind the period of each function and use the language oftransformations to describe how the graphs are related.
(a)1 siny x
(b) 2 2sin3
xy
(c) 3 3sin 2y x
The graph of y is a horizontalstretch of the graph of y by 3, avertical stretch by 2, and areflection across the x-axis,performed in any order.
2
1
![Page 9: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/9.jpg)
TransformationsFind the period of each function and use the language oftransformations to describe how the graphs are related.
(a)1 siny x
(b) 2 2sin3
xy
(c) 3 3sin 2y x
The graph of y is a horizontalshrink of the graph of y by 1/2, avertical stretch by 3, and areflection across the y-axis,performed in any order.
3
1
Confirm these answers Confirm these answers graphically!!!graphically!!!
![Page 10: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/10.jpg)
Definition: Frequency of a Sinusoid
sin[ ( )]f x a b x h k The frequency of the sinusoid
is 2b
Similarly, the frequency of
cos[ ( )]f x a b x h k is 2b Graphically, the frequency is the number of complete cyclesthe wave completes in a unit interval.
Reciprocal of the period!!!
![Page 11: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/11.jpg)
TransformationsFind the frequency of the given function, and interpret itsmeaning graphically.
24sin
3
xf x
Sketch the graph of the function in by 2 ,2 4,4
Frequency:2 3 1
2 3
Period: 3The graph completes 1 full cycleper interval of length .3
Interpretation:
![Page 12: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/12.jpg)
TransformationsHow does the graph of differ from thegraph of ?
y f x c y f x
A translation to the left by c units when c > 0
New Terminology: When applied to sinusoids, wesay that the wave undergoes a phase shift of –c.
![Page 13: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/13.jpg)
TransformationsWrite the cosine function as a phase shift of the sine function.
cos sin 2x x Write the sine function as a phase shift of the cosine function.
sin cos 2x x
Confirm these answers graphically!!!Confirm these answers graphically!!!
![Page 14: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/14.jpg)
Finally, a couple of whiteboard problems
Find the amplitude of the function and use the language oftransformations to describe how the graph of the function isrelated to the graph of the sine function.
2siny x1. Amplitude 2; Vertical stretch by 2Amplitude 2; Vertical stretch by 2
4siny x3. Amplitude 4; Vertical stretch by 4,Amplitude 4; Vertical stretch by 4,Reflect across Reflect across xx-axis-axis
![Page 15: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/15.jpg)
More whiteboard…Find the period of the function and use the language oftransformations to describe how the graph of the function isrelated to the graph of the cosine function.
cos3y x7. Period ; Horizontal shrinkPeriod ; Horizontal shrinkby 1/3by 1/3
cos 7y x 9. Period ; HorizontalPeriod ; Horizontalshrink by 1/7, Reflect acrossshrink by 1/7, Reflect acrossyy-axis-axis
2 3
2 7
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Reminder: Graphs of Sinusoids
The graphs of these functions have the following characteristics:
siny a b x h k cosy a b x h k
0, 0a b
Amplitude = a Period =2
b
Frequency =
2
b
A phase shift of h A vertical translation of k
![Page 17: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/17.jpg)
Guided PracticeGraph one period of the given function by hand.
2.5siny x Amplitude = 2.5 Period = 2
by , 3,3
![Page 18: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/18.jpg)
Guided PracticeGraph one period of the given function by hand.
4cosy x Amplitude = 4 Period = 2
by , 4,4
![Page 19: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/19.jpg)
WhiteboardGraph three periods of the given function by hand.
3cos 2y x Amplitude = 3 Period = 4
by 6 ,6 3,3
![Page 20: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/20.jpg)
WhiteboardGraph three periods of the given function by hand.
20sin 4y x Amplitude = 20 Period =2
by 3 4,3 4 30,30
![Page 21: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/21.jpg)
WhiteboardGraph three periods of the given function by hand.
8cos5y x Amplitude = 8 Period =2
5
by 3 5,3 5 9,9
![Page 22: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/22.jpg)
Guided PracticeIdentify the maximum and minimum values and the zeros of thegiven function in the interval no calculator! 2 ,2
Maximum:
3cos2
xy
3 At 0
Minimum: 3 At 2Zeros:
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WhiteboardIdentify the maximum and minimum values and the zeros of thegiven function in the interval no calculator! 2 ,2
Maximum:
0.5siny x1
2At
3,
2 2
Minimum:1
2 At
2
and
3
2
Zeros: 0, , 2
![Page 24: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/24.jpg)
WhiteboardState the amplitude and period of the given sinusoid, and(relative to the basic function) the phase shift and verticaltranslation.
3.5sin 2 12
y x
Amplitude:3.5 Period: Phase Shift:4
Vertical Translation: 1 unit down
3.5sin 2 14
x
![Page 25: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/25.jpg)
WhiteboardState the amplitude and period of the given sinusoid, and(relative to the basic function) the phase shift and verticaltranslation. 2 3
cos 73 4
xy
Amplitude:2
3Period: 8 Phase Shift: 3
Vertical Translation: 7 units up
2 1cos 3 73 4
x
![Page 26: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/26.jpg)
Practice ProblemsPractice ProblemsConstruct a sinusoid with period and amplitude 6 that goesthrough (2,0).
5
10b
siny a b x h k
2
5b
First, solve for b:
Either will work!!!
6a Find the amplitude:
6a Let’s just take the positive value again.
To pass through (2,0), we need a phaseshift of 2 h = –2
6sin 10 2 6sin 10 20y x x
![Page 27: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/27.jpg)
Practice ProblemsPractice ProblemsConstruct a sinusoid y = f(x) that rises from a minimum value ofy = 5 at x = 0 to a maximum value of y = 25 at x = 32.
First, sketch a graph of this sinusoid…
25 510
2a
Amplitude is half the height:
The period is 64: 264
b
32b
We need a function whose minimum is at x = 0. We could shiftthe sine function horizontally, but it’s easier to simply reflect thecosine function……by letting a = –10
![Page 28: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/28.jpg)
Practice ProblemsPractice ProblemsConstruct a sinusoid y = f(x) that rises from a minimum value ofy = 5 at x = 0 to a maximum value of y = 25 at x = 32.
10cos32
y x
But since cosine is aneven function:
This function ranges from –10 to 10, but we need a functionthat ranges from 5 to 25………vertical translation by 15:
10cos32x
10cos 1532
y x
Support this answerSupport this answer
graphically???graphically???
![Page 29: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/29.jpg)
Amplitude: 2, Period: , Point (0,0)
Practice ProblemsPractice ProblemsConstruct a sinusoid with the given information.
3
2sin 2 3y xOne possibility:
![Page 30: Sinusoids and Transformations Sec. 4.4b. Definition: Sinusoid A function is a sinusoid if it can be written in the form where a, b, c, and d are constants.](https://reader034.fdocuments.us/reader034/viewer/2022052701/56649e975503460f94b9a937/html5/thumbnails/30.jpg)
Amplitude: 3.2, Period: , Point (5,0)
Practice ProblemsPractice ProblemsConstruct a sinusoid with the given information.
7
3.2sin 14 5y x One possibility: