Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y =...
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Transcript of Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y =...
![Page 1: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/1.jpg)
Chapter 3Graphing Trigonometric Functions3.1 Basic Graphics3.2 Graphing y = k + A sin Bx
and y = k +A cos Bx3.3 Graphing y = k + A sin (Bx + C)
and y = k +A cos (Bx + C)3.4 Additional Applications3.5 Graphing Combined Forms3.6 Tangent, Cotangent, Secant, and Cosecant Functions Revisited
![Page 2: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/2.jpg)
3.1 Basic Graphs
Graphs of y = sin x and y = cos xGraphs of y = tan x and y = cot xGraphs of y = csc x and y = sec xGraphing with a graphing calculator
![Page 3: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/3.jpg)
y = sin x
![Page 4: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/4.jpg)
y = cos x
![Page 5: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/5.jpg)
y = tan x
![Page 6: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/6.jpg)
Other Graphs
![Page 7: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/7.jpg)
3.2 Graphing y= k + A sin Bx and y = k + A cos Bx
Graphing y = A sin x and y = A cos xGraphing y = sin Bx and y = cos BxGraphing y = A sin Bx and y = A cos BxGraphing y= k + A sin Bx
and y = k + A cos BxApplications
![Page 8: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/8.jpg)
Comparing Amplitudes
Compare the graphs of y = 1/3 sin x
and y = 3 sin xThe effect of A in
y = A sin x is to increase or decrease the y values without affecting the x values.
![Page 9: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/9.jpg)
Comparing Periods
Compare the graphs of y = sin 2x
and y = sin ½ x
The graph shows the change in the period.
![Page 10: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/10.jpg)
Amplitude and Period
For both y = A sin Bx and y = A cos Bx:
Amplitude = |A| Period = 2/B
![Page 11: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/11.jpg)
Vertical Shift
y = -2 + 3 cos 2x, - x 2Find the period, amplitude, and phase shift and
then graph
![Page 12: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/12.jpg)
Period and Frequency
For any periodic phenomenon, if P is the period and f is the frequency, P = 1/f.
![Page 13: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/13.jpg)
3.3 Graphing y = k + A sin (Bx + C)and y = k + A cos (Bx + C)
Graphing y = A sin (Bx + C)
and y = A cos (Bx + C)Graphing y = k + A sin (Bx + C)
and y = k + A cos (Bx + C)Finding the equation for the graph of a
simple harmonic motion
![Page 14: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/14.jpg)
Finding Period and Phase Shift
y = A sin (Bx + C) and y = A cos (Bx + C) These have the same general shape as
y = A sin Bx
and y = A cos Bx translated horizontally. To find the translation:
x = -C/B (phase shift)
and x = -C/B + 2/B
![Page 15: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/15.jpg)
Phase shift and Period
Find the period and phase shift of
y = sin(2x + /2)
The period is .The phase shift is –/4. 4
22
02
2
x
x
x
![Page 16: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/16.jpg)
Steps for Graphing
![Page 17: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/17.jpg)
3.4 Additional Applications
Modeling electric currentModeling light and other electromagnetic
wavesModeling water wavesSimple and damped harmonic motion:
resonance
![Page 18: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/18.jpg)
Alternating Current Generator
I = 35 sin (40t – 10) (current) Amplitude = 35 Phase shift:
40t = 10t = ¼
Frequency = 1/Period = 20 Hz Period = 1/20
![Page 19: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/19.jpg)
Electromagnetic Waves
E = A sin 2(vt – r/)t = time, r = distance from the source, is the
wavelength, v is the frequency
![Page 20: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/20.jpg)
Water Waves
y = A sin 2p(f1t – r/)
t = time, r = distance from the source, is the wavelength, f1 is the frequency
![Page 21: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/21.jpg)
Damped Harmonic Motion
Y = (1/t)sin (/2)t, 1 t 8First, graph y = 1/t.Then, graph y = sin(t/2) keeping high and low
points within the envelope.
![Page 22: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/22.jpg)
3.6 Tangent, Cotangent, Secant, and Cosecant Functions Revisited
Graphing y = A tan (Bx + C) and
y = cot (Bx + c)Graphing y = A sec (Bx + C) and
y = csc (Bx + c)
![Page 23: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/23.jpg)
y = tan x
![Page 24: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/24.jpg)
y = cot x
![Page 25: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/25.jpg)
y = csc x
![Page 26: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/26.jpg)
Y = sec x
![Page 27: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/27.jpg)
Graphing y = A tan (Bx + C)
Y = 3 tan (/2(x) + /4), -7/2 x 5/2Phase shift = -1/2Period = 2Asymptotes at -7/2, -3/2, ½, and 5/2
![Page 28: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/28.jpg)
Graph of y = sec x
Graph y = 5 sec (1/2(x) + for -7 x 3.
![Page 29: Chapter 3 Graphing Trigonometric Functions 3.1 Basic Graphics 3.2 Graphing y = k + A sin Bx and y = k +A cos Bx 3.3 Graphing y = k + A sin (Bx + C) and.](https://reader030.fdocuments.us/reader030/viewer/2022032703/56649cf95503460f949ca3f9/html5/thumbnails/29.jpg)
Graphing y = A csc(Bx + C)
Graph y = 2 csc (/2(x) = ) for -2 < x < 10