chapter 3 graphing trigonometric functions 3.1 basic graphics 3.2 graphing y = k + a sin bx and y =...
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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
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
y = sin x
y = cos x
y = tan x
Other Graphs
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
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.
Comparing Periods
Compare the graphs of y = sin 2x
and y = sin ½ x
The graph shows the change in the period.
Amplitude and Period
For both y = A sin Bx and y = A cos Bx:
Amplitude = |A| Period = 2/B
Vertical Shift
y = -2 + 3 cos 2x, - x 2Find the period, amplitude, and phase shift and
then graph
Period and Frequency
For any periodic phenomenon, if P is the period and f is the frequency, P = 1/f.
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
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
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
Steps for Graphing
3.4 Additional Applications
Modeling electric currentModeling light and other electromagnetic
wavesModeling water wavesSimple and damped harmonic motion:
resonance
Alternating Current Generator
I = 35 sin (40t – 10) (current) Amplitude = 35 Phase shift:
40t = 10t = ¼
Frequency = 1/Period = 20 Hz Period = 1/20
Electromagnetic Waves
E = A sin 2(vt – r/)t = time, r = distance from the source, is the
wavelength, v is the frequency
Water Waves
y = A sin 2p(f1t – r/)
t = time, r = distance from the source, is the wavelength, f1 is the frequency
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.
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)
y = tan x
y = cot x
y = csc x
Y = sec x
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
Graph of y = sec x
Graph y = 5 sec (1/2(x) + for -7 x 3.
Graphing y = A csc(Bx + C)
Graph y = 2 csc (/2(x) = ) for -2 < x < 10