1. 4.5 Sinusoidal Graphs Sketching and Writing Equations

2. Sinusoidal Graphs Periodic Function Graphs of sine and cosine functions. Functions whose graphs have a repeating pattern. Period The horizontal length of each cycle in a periodic graph. Midline The horizontal line halfway between a sinusoid’s max/min values.

3. The Graph of y = sin x x y

4. Characteristics of the Sine Function The domain is the set of all real numbers. The range consists of all real numbers from -1 to 1, inclusive. The sine function is an odd function (symmetric with respect to the origin). The sine function has a period of 2π Midline is the x-axis

5. Period 3 2 1 -2 -3 y = sinx 1 period pattern starts Period= 2  pattern starts 1 period

6. The Graph of y = cos x x y

7. Characteristics of the Cosine Function The domain is the set of all real numbers. The range consists of all real numbers from -1 to 1, inclusive. The cosine function is an even function (symmetric with respect to the y-axis). The cosine function has a period of 2π Midline is the x-axis

8. Period 3 2 1 -2 -3 1 period y = cosx pattern starts Period = 2  pattern starts 1 period

9. Period 3 2 1 -2 -3 1 period pattern starts Period= 

10. Period 3 2 1 -2 -3 1 period pattern starts Period = 33 2

11. Amplitude Half the distance between the maximum and minimum of a sinusoidal graph.

12. Amplitude 3 2 1 -2 -3 Amplitude = 1 Amplitude max. min.

13. Amplitude 3 2 1 -2 -3 max. min. Amplitude Amplitude = 2

14. Amplitude 3 2 1 -2 -3 max. min. Amplitude Amplitude = 1

15. Amplitude 3 2 1 -2 -3 max. min. Amplitude Amplitude = 2 33

16. Sinusoidal Graphs Amplitude = Period = General Form of Equations

17. Sinusoidal Graphs y = 3sin4x Amplitude = Period = y = -4cos5x Amplitude = Period =

18. Writing Equations Answer these questions first: 1.Is it a “sin” or “cos” graph? 2.What is the amplitude? 3.What is the period? 4.Is the graph reflected? 5.Is it shifted up/down? 6.Is it shifted left/right?

19. Writing Equations b = Write an equation: cosine graph amplitude = 4 Period =  a =4 b == 2

20. Writing Equations b = Write an equation: sine graph amplitude = 3 Period = 4 a =3 b =

21. Writing Equations b = Write an equation: sine graph amplitude = 2 Period = 3 Reflected over x-axis a =±2±2 b =

22. Writing Equations b = a = b = Write an equation: cosine graph amplitude = ½ Period = Reflected over x-axis Invert and multiply!

23. Writing Equations Write an Equation Sine graph Amplitude = 5 Period = Shifted up 2 a = b = Changing the midline causes vertical shifts

24. Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude = b = y = sin2x “sin”  1 a =1

25. Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude b = y = 3sinx “sin” 22 = 3 a =3

26. Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude = “cos” b = 1 a =1

27. Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude = “cos” b = 2 33 Reflected over x-axis! a = -2

28. Sinusoidal Graphs 3 2 1 -2 -3 Period = Amplitude = “cos” b = 3 6 Reflected over x-axis! a =-3

29. Graphing Sinusoidal Functions y = 3sin4x 3 2 1 -2 -3

30. Graphing Sinusoidal Functions y = 2sin  x 3 2 1 -2 -3

31. Graphing Sinusoidal Functions y = -3sin2x 3 2 1 -2 -3

32. Graphing Sinusoidal Functions 3 2 1 -2 -3

33. Graphing Sinusoidal Functions 3 2 1 -2 -3

34. Graphing Sinusoidal Functions 3 2 1 -2 -3

35. Phase Shift Horizontal translations of sine and cosine graphs. Left Right Phase Shift

36. Amp =3 T = P.S. = right

37. Phase Shift Amp =2 T = P.S. = left

38. Phase Shift “Cosine” Amplitude = 5 Period =  Phase shift =

39. Phase Shift “sine” Amplitude = 3 Period = 4 Phase shift =