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Sinusoidal Models (modeling with the sine/cosine functions)

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).

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Sinusoidal Models (modeling with the sine/cosine functions)

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  1. Sinusoidal Models(modeling with the sine/cosine functions) Fraction of the Moon Illuminated at Midnightevery 6 days from January to March 1999 Cyclical NaturePeriodicOscillation

  2. 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 y = k*sin(x) pull them out and squeeze them together y = sin(k*x) move them up and move them down y = sin(x) + k move them left and move them right y = sin(x-k)

  3. stretch them up and squash them down y = Asin(x) y = 1sin(x)Period: 2πMidline: y = 0Amplitude: 1

  4. 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

  5. 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

  6. Sinusoidal Models(modeling with the sine/cosine functions) In the formula f(x) = Asin(x), A is the amplitude of the sine curve.

  7. pull them out and squeeze them together y = sin(Bx) y = sin(1x)Period: 2πMidline: y = 0Amplitude: 1

  8. pull them out and squeeze them together y = sin(Bx) y = 1sin(x)Period: 2πMidline: y = 0Amplitude: 1 y = sin(4x)Period: π/2Midline: y = 0Amplitude: 1

  9. pull them out and squeeze them together 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

  10. 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.

  11. move them up and move them down y = sin(x) + D y = sin(x)Period: 2πMidline: y = 0Amplitude: 1 y = sin(x)+2Period: 2πMidline: y = 2Amplitude: 1 y = sin(x)-1Period: 2πMidline: y = -1Amplitude: 1

  12. 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 .

  13. move them left and move them right y = sin(x-C) y = sin(x)Period: 2πMidline: y = 0Amplitude: 1Phase Shift: none y = sin(x-π)Period: 2πMidline: y = 0Amplitude: 1Phase Shift: π y = sin(x+π/2)Period: 2πMidline: y = 0Amplitude: 1Phase Shift: -π/2

  14. 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

  15. 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

  16. 5/311 f(t) = sin(t) f(t) = sin(3t) f(t) = sin(3t – π/4)

  17. 6&7/311 f(t) = -3sin(0.5t) f(t) = -3sin(0.5(t+1)) f(t) = -3sin(0.5t+1)

  18. 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

  19. 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

  20. More Practice #31, #33, #41, #43, #45 33/329 amplitude 1, period 6, horizontal shift 2 left by hand graph Maple graph

  21. More Practice #31, #33, #41, #43, #45 41/330 write a sine or cosine formula that could represent the given graph

  22. More Practice #31, #33, #41, #43, #45 43/330 write a sine or cosine formula that could represent the given graph

  23. More Practice #31, #33, #41, #43, #45 45/330 write a sine or cosine formula that could represent the given graph

  24. Homework page328 #31-#35, #41-#46 TURN IN: #32, #34, #42, #44, #46 Check your formulas using a Maple graph.

  25. 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

  26. 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

  27. 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

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