Applied Math 40S May 26, 2008
-
Upload
darren-kuropatwa -
Category
Technology
-
view
1.503 -
download
2
description
Transcript of Applied Math 40S May 26, 2008
![Page 1: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/1.jpg)
Applications of Periodic Functions
orBugs On Wheels
Suicidal Shield Bug by flickr user ChinchillaVilla
![Page 2: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/2.jpg)
Properties and Transformations of the sine function ...
Let's look at some graphs ...http://fooplot.com
ƒ(x) = AsinB(x - C) + D
ƒ(x) = Asin(Bx - c) + D
![Page 3: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/3.jpg)
D is the sinusoidal axis, average value of the function, or the vertical shift.
The Role of Parameter D
D < 0 the graph shifts down D units.D > 0 the graph shifts up D units.
ƒ(x) = AsinB(x - C) + D
![Page 4: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/4.jpg)
![Page 5: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/5.jpg)
The amplitude is the absolute value of A; |A|. It is the distance from the sinusoidal axis to a maximum (or minimum). If it is negative, the graph is reflected (flips) over the sinusoidal axis.
The Role of Parameter A ƒ(x) = AsinB(x - C) + D
![Page 6: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/6.jpg)
B is not the period; it determines the period according to this relation: The Role of Parameter B
or
ƒ(x) = AsinB(x - C) + D
![Page 7: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/7.jpg)
C is called the phase shift, or horizontal shift, of the graph.
The Role of Parameter C ƒ(x) = AsinB(x - C) + D
WATCH THE SIGN OF C
when C > 0 the graph shifts right
when C < 0 the graph shifts left
ƒ(x) = AsinB(x - C) + D
ƒ(x) = asin(bx - c) + dc = BC
![Page 8: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/8.jpg)
In general form, the equation and graph of the basic sine function is:
ƒ(x) = AsinB(x - C) + D
In general form, the equation and graph of the basic cosine function is:
ƒ(x) = AcosB(x - C) + D
2π
2π
-2π
-2π
-π
-π π
πSince these graphs are so similar (they differ only by a "phase shift" of π/2 units) we will limit our study to the sine function.
The "starting point."
The "starting point."
Note that your calculator displays: ƒ(x) = asin(bx - c) + d
Which is equivalent to: ƒ(x) = AsinB(x - c/b) + D
A=1, B=1, C=0, D=0
A=1, B=1, C=0, D=0
![Page 9: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/9.jpg)
How many revolutions (in radians and degrees) are illustrated in each graph? How many periods are illustrated in each graph?
Periods = Radians Rotated = Degrees Rotated =
Periods = Radians Rotated = Degrees Rotated =
Periods = Radians Rotated = Degrees Rotated =
HOMEWORK
![Page 10: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/10.jpg)
Determine approximate values for the parameters 'a', 'b', 'c', and 'd' from the graphs, and then write the equations of each graph as a sinusoidal function in the form: y = a sin b(x + c) + d HOMEWORK
ƒ(x) = AsinB(x - C) + D
![Page 11: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/11.jpg)
Determine approximate values for the parameters 'a', 'b', 'c', and 'd' from the graphs, and then write the equations of each graph as a sinusoidal function in the form: y = a sin b(x + c) + d HOMEWORK
ƒ(x) = AsinB(x - C) + D
![Page 12: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/12.jpg)
State the amplitude, period, horizontal shift, and vertical shift for each of the following:
amplitude: period: horizontal shift:vertical shift:
amplitude: period: horizontal shift:vertical shift:
HOMEWORK
![Page 13: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/13.jpg)
State the amplitude, period, horizontal shift, and vertical shift for each of the following:
amplitude: period: horizontal shift:vertical shift:
amplitude: period: horizontal shift:vertical shift:
HOMEWORK
![Page 14: Applied Math 40S May 26, 2008](https://reader034.fdocuments.us/reader034/viewer/2022042606/547d1d875806b5d13f8b4826/html5/thumbnails/14.jpg)
Enter the values into your calculator, and use a sinusoidal regression to determine the equation. Round the values of the parameters to one decimal place.
x -1 -0.5 0 0.5 1 1.5 2 2.5 y 1 -2.6 -5.6 -5.4 -2 1.4 1.6 -1.4
HOMEWORK