1. Feb 11, 2011 The transformed trigonometric functions

2. f(x) = a sin b(x – h) + k Recall which is which in the rule:

3. Match the parameters to the number: a b h k

4. a b h k 5 7 4 1

5. Which is affected by parameter a? Amplitude Period Frequency l.o.o. a = 1

6. Which is affected by parameter a? Amplitude Period Frequency l.o.o. a = 2

7. Which is affected by parameter a? Amplitude Period Frequency l.o.o. a = 3

8. Which is affected by parameter a? Amplitude Period Frequency l.o.o.

9. In fact, parameter a = amplitude Amplitude Period Frequency l.o.o.

10. What would be the amplitude: y = 2 cos x y = 8 sin 2x y = -3 cos x y = 4 sin 9x - 2

11. What would be the amplitude: y = 2 cos x y = 8 sin 2x y = -3 cos x y = 2.4 sin 9x - 2 amplitude = 2 amplitude = 8 amplitude = 3 amplitude = 2.4

12. What would be the value of a in the rule?

13. a = 5

14. What would be the value of a in the rule?

15. a = 4

16. What would be the value of a in the rule? a = 4

17. Another way to find amplitude: Amplitude = half the distance between the Max and min values = (M – m)  2 = (2 - -6)  2 = 8  2 = 4

18. Another way to find amplitude: Amplitude = half the distance between the Max and min values = (M – m)  2 = (2 - -6)  2 = 8  2 = 4 2 -6

19. What would be the value of a in the rule?

20. a = 1 Amplitude = half the distance between the Max and min values = (M – m)  2 = (2 - 0)  2 = 2  2 = 1

21. In general then: For f(x) = a sin b(x – h) + k OR: f(x) = a cos b(x – h) + k Amplitude =

22. In general then: For f(x) = a sin b(x – h) + k OR: f(x) = a cos b(x – h) + k Amplitude = |a|

23. In general then: For f(x) = a sin b(x – h) + k OR: f(x) = a cos b(x – h) + k Amplitude = |a|

24. Which is affected by parameter b? Amplitude Period Frequency l.o.o. b = 1

25. Which is affected by parameter b? Amplitude Period Frequency l.o.o. b = 2

26. Which is affected by parameter b? Amplitude Period Frequency l.o.o. b = 4

27. Which is affected by parameter b? Amplitude Period Frequency l.o.o.

28. Which is affected by parameter b? Amplitude Period Frequency l.o.o. 4 cycles

29. Which is affected by parameter b? Amplitude Period Frequency l.o.o.

30. In fact, b = frequency Amplitude Period Frequency = 4 = b l.o.o. y = sin 4x

31. What would be the frequency: y = cos 4x y = 8 sin 2x y = -3 cos  (x + 1) -2 y = 2.4 sin (-9x) - 2

32. What would be the frequency: y = cos 4x y = 8 sin 2x y = -3 cos  (x + 1) -2 y = 2.4 sin (-9x) - 2 frequency = 4 frequency = 2 frequency =  frequency = 9

33. What would be the value of b in the rule?

34. b = 1

35. What would be the value of b in the rule?

36. b = 3

37. What would be the value of b in the rule?

38. b = 0.5

39. In general then: For f(x) = a sin b(x – h) + k OR: f(x) = a cos b(x – h) + k Frequency =

40. In general then: For f(x) = a sin b(x – h) + k OR: f(x) = a cos b(x – h) + k Frequency = |b|

41. And if 4 cycles have a total width of 2 .......then one of those cycles must have a width of... Amplitude Period Frequency l.o.o. y = sin 4x

42. And if 4 cycles have a total width of 2 .......then one of those cycles must have a width of... Amplitude Period Frequency l.o.o. y = sin 4x ?

43. Amplitude Period = Frequency l.o.o. y = sin 4x And if 4 cycles have a total width of 2 .......then one of those cycles must have a width of...

44. Amplitude Period = Frequency l.o.o. y = sin 4x And if 4 cycles have a total width of 2 .......then one of those cycles must have a width of...

45. Amplitude Period = Frequency l.o.o. y = sin 4x In fact, period =

46. Amplitude Period = Frequency l.o.o. y = sin 4x In fact, period =

47. What would be the period: y = cos 4x y = 8 sin 2x y = -3 cos  (x + 1) -2 y = 2.4 sin (-9x) - 2 period =

48. What would be the period: y = cos 4x y = 8 sin 2x y = -3 cos  (x + 1) -2 y = 2.4 sin (-9x) - 2 period =

49. In general then: For f(x) = a sin b(x – h) + k OR: f(x) = a cos b(x – h) + k Frequency = |b| Period =

50. Which is affected by parameter h? Amplitude Period Frequency l.o.o. h = 0

51. Which is affected by parameter h? Amplitude Period Frequency l.o.o. h =.3 

52. Which is affected by parameter h? Amplitude Period Frequency l.o.o. h =.5 

53. Which is affected by parameter h? Amplitude Period Frequency l.o.o.

54. But h does shift horizontally...and this shift has a special name: Phase shift Amplitude Period Frequency l.o.o.

55. What would be the phase shift: y = cos 4x + 1 y = 8 sin 2(x -  ) -3 y = -3 cos  (x + 1) -2 y = 2.4 sin (2x +  ) phase shift =

56. What would be the phase shift: y = cos 4x + 1 y = 8 sin 2(x -  ) -3 y = -3 cos  (x + 1) -2 y = 2.4 sin (2x +  ) phase shift = 0 phase shift =  phase shift = -1 phase shift =

57. What would be the value of h in the rule?

58. If we consider this to be a sine function, h =

59. What would be the value of h in the rule? If we consider this to be a sine function, h = Snake is beginning here

60. What would be the value of h in the rule? If we consider this to be a sine function, h = Which is  /2 to the right of where it usually begins

61. What would be the value of h in the rule? If we consider this to be a sine function, h = In the rule, you would see:

62. What would be the value of h in the rule? If we consider this to be a cos function, h =

63. What would be the value of h in the rule? If we consider this to be a cos function, h = Tulip is beginning here

64. What would be the value of h in the rule? If we consider this to be a cos function, h = Which is  to the right of where it usually begins

65. What would be the value of h in the rule? If we consider this to be a cos function, h = Which is  to the right of where it usually begins

66. What would be the value of h in the rule? If we consider this to be a cos function, h = In the rule, you would see: (x -  )

67. What would be the value of h in the rule?

68. If considered as a sine function, h =

69. If considered as a cos function, h =

70. What would be the value of h in the rule?

71. As a cos: h = 0

72. Which is affected by parameter k? Amplitude Period Frequency l.o.o. k = 0

73. Which is affected by parameter k? Amplitude Period Frequency l.o.o. k = 1

74. Which is affected by parameter k? Amplitude Period Frequency l.o.o. k = 2

75. Which is affected by parameter k? Amplitude Period Frequency l.o.o.

76. In fact, l.o.o. has equation: y = k Amplitude Period Frequency l.o.o.

77. What would be the l.o.o.: y = cos 4x + 1 y = 8 sin 2(x -  ) - 3 y = -3 cos  (x + 1) - 2 y = 2.4 sin (2x +  )

78. What would be the l.o.o.: y = cos 4x + 1 y = 8 sin 2(x -  ) - 3 y = -3 cos  (x + 1) - 2 y = 2.4 sin (2x +  ) l.o.o.: y = 1 l.o.o.: y = -3 l.o.o.: y = -2 l.o.o.: y = 0

79. What would be the value of k in the rule?

80. k = -1

81. Another way to find k: k = the number halfway between the Max and min values = (M + m)  2 = (1 + -3)  2 = -2  2 = -1

82. Another way to find k: k = the number halfway between the Max and min values = (M + m)  2 = (1 + -3)  2 = -2  2 = -1

83. What would be the value of k in the rule?

84. k = the number halfway between the Max and min values = (M + m)  2 = (0 + -2)  2 = -2  2 = -1

85. In general then: For f(x) = a sin b(x – h) + k OR: f(x) = a cos b(x – h) + k l.o.o. is the line y = k

86. And another thing.... For f(x) = a sin b(x – h) + k OR: f(x) = a cos b(x – h) + k Max = k + amplitude min = k - amplitude

87. And another thing.... For f(x) = a sin b(x – h) + k OR: f(x) = a cos b(x – h) + k Max = k + amplitude min = k - amplitude

88. y = 3 sin 2x - 1

89. y = -1

90. y = 3 sin 2x - 1 y = -1

91. y = 3 sin 2x - 1 22

92. 22

93. P = 2  /2 = 

94. Find the rule:

95. y = 2 cos x

96. Find the rule:

97. y = 3 sin x

98. Find the rule:

99. y = 3 sin 2x

100. Find the rule:

101. y = 3 sin 2x - 1

102. Find the rule:

103. y = 2 sin 3(x -  /4) + 1

104. y = 2 cos 3(x +  /4) + 1

105. Hwk: Blog Three gizmos: –Cosine function –Sine function –Translating and scaling Sine and Cosine functions – Activity A Carousel: –p. 253 #6, 9ab, 10abd, 19 –p. 263 #6, 9, 10