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Sinusoidal Models (modeling with the sine/cosine functions) Fraction of the Moon Illuminated at Midnight every 6 days from January to March 1999 Cyclical Nature Periodic Oscillation
Sinusoidal Models(modeling with the sine/cosine functions)
Fraction of the Moon Illuminated at Midnightevery 6 days from January to March 1999
Cyclical NaturePeriodic
Oscillation
Sinusoidal Models(modeling with the sine/cosine functions)
To use sine and cosine functions for modeling, we must be able to:
stretch them up and squash them down
pull them out and squeeze them together
move them up and move them down
move them left and move them right
y = k*sin(x)
y = sin(k*x)
y = sin(x) + k
y = sin(x-k)
stretch them up and squash them down
y = Asin(x)
y = 1sin(x)Period: 2πMidline: y = 0Amplitude: 1
y = 1sin(x)Period: 2πMidline: y = 0Amplitude: 1
y = 3sin(x)Period: 2πMidline: y = 0Amplitude: 2
stretch them up and squash them down
y = Asin(x)
y = 1sin(x)Period: 2πMidline: y = 0Amplitude: 1
y = 3sin(x)Period: 2πMidline: y = 0Amplitude: 2
y = -0.5sin(x)Period: 2πMidline: y = 0Amplitude: 0.5
stretch them up and squash them down
y = Asin(x)
Sinusoidal Models(modeling with the sine/cosine functions)
In the formula f(x) = Asin(x), A is the amplitude of the sine curve.
pull them out and squeeze them together
y = sin(Bx)
y = sin(1x)Period: 2πMidline: y = 0Amplitude: 1
y = 1sin(x)Period: 2πMidline: y = 0Amplitude: 1
y = sin(4x)Period: π/2Midline: y = 0Amplitude: 1
pull them out and squeeze them togetherpull them out and squeeze them together
y = sin(Bx)y = sin(Bx)
y = 1sin(x)Period: 2πMidline: y = 0Amplitude: 1
y = sin(4x)Period: π/2Midline: y = 0Amplitude: 1
y = sin(0.5x)Period: 4πMidline: y = 0Amplitude: 1
pull them out and squeeze them togetherpull them out and squeeze them together
y = sin(Bx)y = sin(Bx)
Sinusoidal ModelsSinusoidal Models(modeling with the sine/cosine functions)(modeling with the sine/cosine functions)
In the formula f(x) = Asin(x), the amplitude of the curve is A.In the formula f(x) = Asin(x), the amplitude of the curve is A.
In the formula f(x) = sin(Bx), the period of the curve is 2π/B.
y = sin(x)Period: 2πMidline: y = 0Amplitude: 1
y = sin(x)+2Period: 2πMidline: y = 2Amplitude: 1
y = sin(x)-1Period: 2πMidline: y = -1Amplitude: 1
move them up and move them down
y = sin(x) + D
Sinusoidal Models(modeling with the sine/cosine functions)
In the formula f(x) = Asin(x), the amplitude of the curve is A.
In the formula f(x) = sin(Bx), the period of the curve is 2π/B.
In the formula f(x) = sin(x) + D, the midline of the curve is y = D .
y = sin(x)Period: 2πMidline: y = 0Amplitude: 1Phase Shift: none
y = sin(x+π/2)Period: 2πMidline: y = 0Amplitude: 1Phase Shift: -π/2
y = sin(x-π)Period: 2πMidline: y = 0Amplitude: 1Phase Shift: π
move them left and move them right
y = sin(x-C)
Sinusoidal Models(modeling with the sine/cosine functions)
In the formula f(x) = Asin(x), the amplitude of the curve is A.
In the formula f(x) = sin(Bx), the period of the curve is 2π/B.
In the formula f(x) = sin(x) + D, the midline of the curve is y = D .
In the formula f(x) = sin(x-C), the phase shift of the curve is C
Sinusoidal Models(modeling with the sine/cosine functions)
f(x) = Asin(B(x-C)) + D
The amplitude of the curve is A.
The period of the curve is 2π/B.
The midline of the curve is y = D .
The phase shift of the curve is C.
CYU 6.8/311
f(t) = sin(t)
f(t) = sin(3t)
f(t) = sin(3t – π/4)
5/311
f(t) = -3sin(0.5t)
f(t) = -3sin(0.5(t+1))
f(t) = -3sin(0.5t+1)
6&7/311
Sinusoidal Models(modeling with the sine/cosine functions)
f(x) = Asin(B(x-C)) + D
The amplitude of the curve is A.
The period of the curve is 2π/B.
The midline of the curve is y = D .
The phase shift of the curve is C.
More Practice
#31, #33, #41, #43, #45
More Practice #31, #33, #41, #43, #45
31’/329 amplitude 3, period π/4, vertical shift 2 down
f(x) = 3sin(8x) - 2
by hand graph
Maple graph
More Practice #31, #33, #41, #43, #45
33/329 amplitude 1, period 6, horizontal shift 2 left
by hand graph
Maple graph
2f(x) = 1sin( ( 2)) sin( )
3 3 3x x
f(x) = sin( )3x
f(x) = sin( ( 2))
3x
More Practice #31, #33, #41, #43, #45
41/330 write a sine or cosine formula that could represent the given graph
More Practice #31, #33, #41, #43, #45
43/330 write a sine or cosine formula that could represent the given graph
More Practice #31, #33, #41, #43, #45
45/330 write a sine or cosine formula that could represent the given graph
Homework
page328 #31-#35, #41-#46
TURN IN: #32, #34, #42, #44, #46
Check your formulas using a Maple graph.
Fraction of the Moon Illuminated at Midnightevery 6 days from January to March 1999
Period is 30, so B = π/15Midline is y = 0.5Amplitude is 0.5.
m(t) = 0.5sin(π/15*(t-C))+0.5
use graph to determine C
Fraction of the Moon Illuminated at Midnightevery 6 days from January to March 1999
Period is 30, so B = π/15Midline is y = 0.5Amplitude is 0.5.
m(t) = 0.5sin(π/15*(t-C))+0.5
use graph to determine C
Fraction of the Moon Illuminated at Midnightevery 6 days from January to March 1999
Period is 30, so B = π/15Midline is y = 0.5Amplitude is 0.5.
C is 6 units left
m(t) = 0.5sin(π/15*(t-6))+0.5
