1. 4.2 – Translations of the Graphs of the Sine and Cosine Functions
Math 150
4.2 – Translations of the Graphs of the Sine and Cosine Functions

2. Here are examples of common transformations on functions (here, the example function is sin 𝑥 ): 𝑦=2 sin 𝑥 stretches vertically by factor of 2 𝑦= sin 2𝑥 shrinks horizontally by factor of 1 2 𝑦= sin 𝑥− 𝜋 3 shifts right by 𝜋 3 𝑦=1+ sin 𝑥 shifts up by 1

3. Here are examples of common transformations on functions (here, the example function is sin 𝑥 ): 𝑦=2 sin 𝑥 stretches vertically by factor of 2 𝑦= sin 2𝑥 shrinks horizontally by factor of 1 2 𝑦= sin 𝑥− 𝜋 3 shifts right by 𝜋 3 𝑦=1+ sin 𝑥 shifts up by 1

4. Here are examples of common transformations on functions (here, the example function is sin 𝑥 ): 𝑦=2 sin 𝑥 stretches vertically by factor of 2 𝑦= sin 2𝑥 shrinks horizontally by factor of 1 2 𝑦= sin 𝑥− 𝜋 3 shifts right by 𝜋 3 𝑦=1+ sin 𝑥 shifts up by 1

5. Here are examples of common transformations on functions (here, the example function is sin 𝑥 ): 𝑦=2 sin 𝑥 stretches vertically by factor of 2 𝑦= sin 2𝑥 shrinks horizontally by factor of 1 2 𝑦= sin 𝑥− 𝜋 3 shifts right by 𝜋 3 𝑦=1+ sin 𝑥 shifts up by 1

6. Here are examples of common transformations on functions (here, the example function is sin 𝑥 ): 𝑦=2 sin 𝑥 stretches vertically by factor of 2 𝑦= sin 2𝑥 shrinks horizontally by factor of 1 2 𝑦= sin 𝑥− 𝜋 3 shifts right by 𝜋 3 𝑦=1+ sin 𝑥 shifts up by 1

7. Transformations of Functions For a function 𝑓(𝑥), and 𝑐>0, 𝑓 𝑥 +𝑐 shifts up 𝑓 𝑥 −𝑐 shifts down 𝑐𝑓(𝑥) stretches vertically (if 𝑐>1), or shrinks vertically (if 0<𝑐<1) −𝑓(𝑥) reflects about 𝑥-axis 𝑓 𝑥−𝑐 shifts right 𝑓 𝑥+𝑐 shifts left 𝑓(𝑐𝑥) stretches horizontally (if 0<𝑐<1), or shrink horizontally (if 𝑐>1) 𝑓(−𝑥) reflects about 𝑦-axis

8. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

9. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

10. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

11. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

12. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

13. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

14. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

15. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

16. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

17. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

18. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

19. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

20. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

21. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

22. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

23. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

24. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

25. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

26. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .

27. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .
In the example above, 𝜋 3 is called the ____________.

28. Ex 1. Graph 𝑦= sin 𝑥− 𝜋 3 .
In the example above, 𝜋 3 is called the ____________.
phase shift

29. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

30. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

31. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

32. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

33. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

34. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

35. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

36. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

37. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

38. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

39. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

40. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

41. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

42. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

43. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

44. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

45. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

46. Ex 2. Graph 𝑦=3 cos 𝑥+ 𝜋 4 and find the amplitude, period, and phase shift.

47. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

48. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

49. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

50. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

51. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

52. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

53. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

54. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

55. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

56. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

57. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

58. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

59. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

60. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

61. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

62. Ex 3. Graph 𝑦=−2 cos 3𝑥+𝜋 and find the amplitude, period, and phase shift.

63. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

64. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

65. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

66. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

67. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

68. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

69. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

70. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

71. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

72. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

73. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

74. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

75. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

76. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

77. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

78. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

79. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.

80. Ex 4. Graph 𝑦=−1+2 sin 4𝑥+𝜋 and find the amplitude, period, and phase shift.