1. 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

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 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)

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

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

5. y = 1sin(x) Period: 2π Midline: y = 0 Amplitude: 1 y = 3sin(x) Period: 2π Midline: y = 0 Amplitude: 2 y = -0.5sin(x) Period: 2π Midline: y = 0 Amplitude: 0.5 stretch them up and squash them down y = Asin(x)

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: 2π Midline: y = 0 Amplitude: 1

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

9. y = 1sin(x) Period: 2π Midline: y = 0 Amplitude: 1 y = sin(4x) Period: π/2 Midline: y = 0 Amplitude: 1 y = sin(0.5x) Period: 4π Midline: y = 0 Amplitude: 1 pull them out and squeeze them together y = sin(Bx)

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

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

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

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

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 Midnight every 6 days from January to March 1999 Period is 30, so B = π/15 Midline is y = 0.5 Amplitude 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 Midnight every 6 days from January to March 1999 Period is 30, so B = π/15 Midline is y = 0.5 Amplitude 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 Midnight every 6 days from January to March 1999 Period is 30, so B = π/15 Midline is y = 0.5 Amplitude is 0.5. C is 6 units left m(t) = 0.5sin(π/15*(t-6))+0.5