1. 4.1 – Graphs of the Sine and Cosine Functions
Math 150
4.1 – Graphs of the Sine and Cosine Functions

2. A periodic function is a function 𝑓 such that 𝑓 𝑥 =𝑓(𝑥+𝑛𝑝) for every real # 𝑥 in the domain of 𝑓, every integer 𝑛, and some positive real number 𝑝. The least possible positive value of 𝑝 is the period of the function.

3. Note: sin 𝑥 and cos 𝑥 are periodic with period _____.

4. Note: sin 𝑥 and cos 𝑥 are periodic with period _____. 𝟐𝝅

5. Let’s try graphing 𝑦= sin 𝑥 .

6. Let’s try graphing 𝑦= sin 𝑥 .

7. Let’s try graphing 𝑦= sin 𝑥 .

8. Let’s try graphing 𝑦= sin 𝑥 .

9. Let’s try graphing 𝑦= sin 𝑥 .

10. Let’s try graphing 𝑦= sin 𝑥 .

11. Let’s try graphing 𝑦= sin 𝑥 .

12. Let’s try graphing 𝑦= sin 𝑥 .

13. Let’s try graphing 𝑦= sin 𝑥 .

14. Let’s try graphing 𝑦= sin 𝑥 .

15. Let’s try graphing 𝑦= sin 𝑥 .

16. Let’s try graphing 𝑦= sin 𝑥 .

17. Let’s try graphing 𝑦= sin 𝑥 .

18. Let’s try graphing 𝑦= sin 𝑥 .

19. Now let’s graph 𝑦= cos 𝑥 .

20. Now let’s graph 𝑦= cos 𝑥 .

21. Now let’s graph 𝑦= cos 𝑥 .

22. Now let’s graph 𝑦= cos 𝑥 .

23. Now let’s graph 𝑦= cos 𝑥 .

24. Now let’s graph 𝑦= cos 𝑥 .

25. Now let’s graph 𝑦= cos 𝑥 .

26. Now let’s graph 𝑦= cos 𝑥 .

27. Now let’s graph 𝑦= cos 𝑥 .

28. Now let’s graph 𝑦= cos 𝑥 .

29. Now let’s graph 𝑦= cos 𝑥 .

30. Now let’s graph 𝑦= cos 𝑥 .

31. Now let’s graph 𝑦= cos 𝑥 .

32. Now let’s graph 𝑦= cos 𝑥 .

33. Ex 1. Graph 𝑦=2 sin 𝑥 .

34. Ex 1. Graph 𝑦=2 sin 𝑥 .

35. Ex 1. Graph 𝑦=2 sin 𝑥 .

36. Ex 1. Graph 𝑦=2 sin 𝑥 .

37. Ex 1. Graph 𝑦=2 sin 𝑥 .

38. Ex 1. Graph 𝑦=2 sin 𝑥 .

39. Ex 1. Graph 𝑦=2 sin 𝑥 .

40. Ex 1. Graph 𝑦=2 sin 𝑥 .

41. Ex 1. Graph 𝑦=2 sin 𝑥 .

42. Ex 1. Graph 𝑦=2 sin 𝑥 .

43. Ex 1. Graph 𝑦=2 sin 𝑥 .

44. Ex 1. Graph 𝑦=2 sin 𝑥 .

45. Note: The amplitude of a periodic function is half the difference between the maximum and minimum values. For 𝑦=𝑎 sin 𝑥 and 𝑦=𝑎 cos 𝑥 the amplitude is 𝑎 .

46. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

47. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

48. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

49. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

50. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

51. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

52. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

53. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

54. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

55. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

56. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

57. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

58. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

59. Ex 2. Graph 𝑦= sin 2𝑥 over a two-period interval.

60. Note: The period of both 𝑦= sin 𝑏𝑥 and 𝑦= cos 𝑏𝑥 is 2𝜋 𝑏 .

61. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

62. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

63. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

64. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

65. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

66. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

67. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

68. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

69. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

70. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

71. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

72. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

73. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.

74. Ex 3. Graph 𝑦=−2 cos 3𝑥 over a 2-period interval.