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# Graphing trigonometric functions

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### Graphing trigonometric functions

1. 1. Graphing Trigonometric Functions Mathematics 4 October 6, 20111 of 45
2. 2. Graphing the function y = sin x:Identify the regions in the cartesian plane corresponding to thequadrants of the unit circle: 2 of 45
3. 3. Graphing the function y = sin x:Identify the regions in the cartesian plane corresponding to thequadrants of the unit circle: 2 of 45
4. 4. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
5. 5. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
6. 6. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
7. 7. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
8. 8. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
9. 9. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
10. 10. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
11. 11. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
12. 12. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
13. 13. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
14. 14. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
15. 15. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
16. 16. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
17. 17. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
18. 18. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
19. 19. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
20. 20. Graphing the function y = sin x:Plotting the sine values of the special angles: 3 of 45
21. 21. Expanding the graph of y = sin x:Graphing beyond [0, 2π]: 4 of 45
22. 22. Properties of the graph of y = sin x:Domain:Range:Zeros:5 of 45
23. 23. Properties of the graph of y = sin x:Domain: RRange: y ∈ [−1, 1]Zeros: {x|x = nπ, n ∈ Z} 6 of 45
24. 24. Properties of the graph of y = sin x:Increasing in the following quadrants:Decreasing in the following quadrants: 7 of 45
25. 25. Properties of the graph of y = sin x:Increasing in the following quadrants: Q1 and Q4Decreasing in the following quadrants: Q2 and Q3 8 of 45
26. 26. Properties of the graph of y = sin x:Amplitude: One-half of the distance from the maximum to theminimum valueThe amplitude of y = sin x is: 9 of 45
27. 27. Properties of the graph of y = sin x:Amplitude: One-half of the distance from the maximum to theminimum valueThe amplitude of y = sin x is: 110 of 45
28. 28. Properties of the graph of y = sin x:Period: The distance from crest-to-crest or trough-to-troughThe period of y = sin x is:11 of 45
29. 29. Properties of the graph of y = sin x:Period: The distance from crest-to-crest or trough-to-troughThe period of y = sin x is 2π.12 of 45
30. 30. Graphing y = sin(x − c):Describe the graph of y = sin(x − π ). 213 of 45
31. 31. Graphing y = sin(x − c):Describe the graph of y = sin(x − π ). 213 of 45
32. 32. Graphing y = sin(x − c):Describe the graph of y = sin(x − π ). 2 πThe graph shifted 2 units to the right.13 of 45
33. 33. Graphing y = cos x:Recall: π cos x = sin −x 2 π = sin − x − 2 π = − sin x − 214 of 45
34. 34. Graphing y = cos x:Graph of y = sin x15 of 45
35. 35. Graphing y = cos x: πGraph of y = sin x − 215 of 45
36. 36. Graphing y = cos x: πGraph of y = − sin x − 215 of 45
37. 37. Graphing y = cos x: πGraph of y = − sin x − 2 = cos x15 of 45
38. 38. Properties of y = cos x:Domain:Range:Zeros:Increasing in:Decreasing in:Amplitude:Period: 16 of 45
39. 39. Properties of y = cos x:Domain: RRange:Zeros:Increasing in:Decreasing in:Amplitude:Period: 16 of 45
40. 40. Properties of y = cos x:Domain: RRange: y ∈ [−1, 1]Zeros:Increasing in:Decreasing in:Amplitude:Period: 16 of 45
41. 41. Properties of y = cos x:Domain: RRange: y ∈ [−1, 1]Zeros: {x|x = nπ , n is an odd integer } 2Increasing in:Decreasing in:Amplitude:Period: 16 of 45
42. 42. Properties of y = cos x:Domain: RRange: y ∈ [−1, 1]Zeros: {x|x = nπ , n is an odd integer } 2Increasing in: Q3 and Q4Decreasing in:Amplitude:Period: 16 of 45
43. 43. Properties of y = cos x:Domain: RRange: y ∈ [−1, 1]Zeros: {x|x = nπ , n is an odd integer } 2Increasing in: Q3 and Q4Decreasing in: Q1 and Q2Amplitude:Period: 16 of 45
44. 44. Properties of y = cos x:Domain: RRange: y ∈ [−1, 1]Zeros: {x|x = nπ , n is an odd integer } 2Increasing in: Q3 and Q4Decreasing in: Q1 and Q2Amplitude: 1Period: 16 of 45
45. 45. Properties of y = cos x:Domain: RRange: y ∈ [−1, 1]Zeros: {x|x = nπ , n is an odd integer } 2Increasing in: Q3 and Q4Decreasing in: Q1 and Q2Amplitude: 1Period: 2π 16 of 45
46. 46. A comparison of y = sin x and y = cos x:Identical properties:Symmetry:17 of 45
47. 47. A comparison of y = sin x and y = cos x:Identical properties: Domain, Range, Amplitude, PeriodSymmetry: y = sin x is symmetric wrt the origin. (Odd function)17 of 45
48. 48. A comparison of y = sin x and y = cos x:Identical properties: Domain, Range, Amplitude, PeriodSymmetry: y = cos x is symmetric wrt the y-axis. (Even function)17 of 45
49. 49. Pick-up quiz: 1 th sheet of paper. 418 of 45
50. 50. Pick-up quiz: 1 th sheet of paper. 41. What is the range of the sine and cosine functions?18 of 45
51. 51. Pick-up quiz: 1 th sheet of paper. 41. What is the range of the sine and cosine functions?2. Which function has its zeros at integer multiples of π?18 of 45
52. 52. Pick-up quiz: 1 th sheet of paper. 41. What is the range of the sine and cosine functions?2. Which function has its zeros at integer multiples of π?3. The cosine function is equivalent to the sine function shifted to the right by this value:18 of 45
53. 53. Pick-up quiz: 1 th sheet of paper. 41. What is the range of the sine and cosine functions?2. Which function has its zeros at integer multiples of π?3. The cosine function is equivalent to the sine function shifted to the right by this value:4. In what quadrant(s) is/are the sine function decreasing?18 of 45
54. 54. Pick-up quiz: 1 th sheet of paper. 41. What is the range of the sine and cosine functions?2. Which function has its zeros at integer multiples of π?3. The cosine function is equivalent to the sine function shifted to the right by this value:4. In what quadrant(s) is/are the sine function decreasing?5. What is the amplitude of the cosine function?18 of 45
55. 55. Graphing sinusoidal functionsThe general form of a sinusoidal function is: f (x) = a sin(b(x − c)) + d or f (x) = a cos(b(x − c)) + dwhere a, b, c, and d modify the basic sine or cosine function.19 of 45
56. 56. Graphing f (x) = a sin xGiven: f (x) = sin xPlot the graph of f (x) = 2 sin x.20 of 45
57. 57. Graphing f (x) = a sin xGiven: f (x) = sin xPlot the graph of f (x) = 2 sin x.20 of 45
58. 58. Graphing f (x) = a sin xGiven: f (x) = sin x 1Plot the graph of f (x) = 2 sin x.21 of 45
59. 59. Graphing f (x) = a sin xGiven: f (x) = sin x 1Plot the graph of f (x) = 2 sin x.21 of 45
60. 60. Graphing f (x) = a sin xGiven: f (x) = sin xPlot the graph of f (x) = − 3 sin x. 222 of 45
61. 61. Graphing f (x) = a sin xGiven: f (x) = sin xPlot the graph of f (x) = − 3 sin x. 222 of 45
62. 62. Graphing f (x) = a sin xSummarize how multiplying a sinusoidal function by a aﬀects thegraph:23 of 45
63. 63. Graphing f (x) = a sin xSummarize how multiplying a sinusoidal function by a aﬀects thegraph:1. |a| > 1 → expands graph vertically23 of 45
64. 64. Graphing f (x) = a sin xSummarize how multiplying a sinusoidal function by a aﬀects thegraph:1. |a| > 1 → expands graph vertically2. |a| < 1 → compresses graph vertically23 of 45
65. 65. Graphing f (x) = a sin xSummarize how multiplying a sinusoidal function by a aﬀects thegraph:1. |a| > 1 → expands graph vertically2. |a| < 1 → compresses graph vertically3. a < 0 → ﬂips the graph vertically23 of 45
66. 66. Graphing f (x) = a sin xSummarize how multiplying a sinusoidal function by a aﬀects thegraph:1. |a| > 1 → expands graph vertically2. |a| < 1 → compresses graph vertically3. a < 0 → ﬂips the graph vertically4. The amplitude of f (x) = a sin x is |a|23 of 45
67. 67. Graphing f (x) = cos(b · x)Given: f (x) = cos xPlot the graph of f (x) = cos(2 · x).24 of 45
68. 68. Graphing f (x) = cos(b · x)Given: f (x) = cos xPlot the graph of f (x) = cos(2 · x).24 of 45
69. 69. Graphing f (x) = cos(b · x)Given: f (x) = cos xPlot the graph of f (x) = cos(2 · x).24 of 45
70. 70. Graphing f (x) = cos(b · x)Given: f (x) = cos xPlot the graph of f (x) = cos( 1 · x). 225 of 45
71. 71. Graphing f (x) = cos(b · x)Given: f (x) = cos xPlot the graph of f (x) = cos( 1 · x). 225 of 45
72. 72. Graphing f (x) = cos(b · x)Given: f (x) = cos xPlot the graph of f (x) = cos( 1 · x). 225 of 45
73. 73. Graphing f (x) = sin(b · x)Given: f (x) = sin xPlot the graph of f (x) = sin(− 4 · x). 326 of 45
74. 74. Graphing f (x) = sin(b · x)Given: f (x) = sin xPlot the graph of f (x) = sin(− 4 · x). 326 of 45
75. 75. Graphing f (x) = sin(b · x)Given: f (x) = sin xPlot the graph of f (x) = sin(− 4 · x). 326 of 45
76. 76. Graphing f (x) = sin(b · x)Given: f (x) = sin xPlot the graph of f (x) = sin(− 4 · x). 326 of 45
77. 77. Graphing f (x) = sin(b · x)Summarize how multiplying the argument of a sinusoidal function byb aﬀects the graph:27 of 45
78. 78. Graphing f (x) = sin(b · x)Summarize how multiplying the argument of a sinusoidal function byb aﬀects the graph:1. |b| > 1 → compresses graph horizontally27 of 45
79. 79. Graphing f (x) = sin(b · x)Summarize how multiplying the argument of a sinusoidal function byb aﬀects the graph:1. |b| > 1 → compresses graph horizontally2. |b| < 1 → expands graph horizontally27 of 45
80. 80. Graphing f (x) = sin(b · x)Summarize how multiplying the argument of a sinusoidal function byb aﬀects the graph:1. |b| > 1 → compresses graph horizontally2. |b| < 1 → expands graph horizontally3. b < 0 → ﬂips the graph horizontally27 of 45
81. 81. Graphing f (x) = sin(b · x)Summarize how multiplying the argument of a sinusoidal function byb aﬀects the graph:1. |b| > 1 → compresses graph horizontally2. |b| < 1 → expands graph horizontally3. b < 0 → ﬂips the graph horizontally 2π4. The period of f (x) = sin(b · x) is |b|27 of 45
82. 82. Graphing f (x) = a · sin(b · x)Identify the amplitude and period, and sketch the graph: 2x1. f (x) = cos 32. g(x) = 4 cos(2π · x)3. h(x) = −2 sin(π · x)4. f (x) = sin(−3x)28 of 45
83. 83. Graphing f (x) = cos(x + c)Given: f (x) = cos xPlot the graph of f (x) = cos(x + π ). 329 of 45
84. 84. Graphing f (x) = cos(x + c)Given: f (x) = cos xPlot the graph of f (x) = cos(x + π ). 329 of 45
85. 85. Graphing f (x) = cos(x + c)Given: f (x) = cos xPlot the graph of f (x) = cos(x + π ). 329 of 45
86. 86. Graphing f (x) = cos(x + c)Given: f (x) = cos x 5πPlot the graph of f (x) = cos(x − 6 ).30 of 45
87. 87. Graphing f (x) = cos(x + c)Given: f (x) = cos x 5πPlot the graph of f (x) = cos(x − 6 ).30 of 45
88. 88. Graphing f (x) = cos(x + c)Given: f (x) = cos x 5πPlot the graph of f (x) = cos(x − 6 ).30 of 45
89. 89. Graphing f (x) = cos(x + c)Summarize how adding c to the argument of a sinusoidal functionaﬀects the graph:31 of 45
90. 90. Graphing f (x) = cos(x + c)Summarize how adding c to the argument of a sinusoidal functionaﬀects the graph:1. f (x + c) → shifts the graph c units to the left31 of 45
91. 91. Graphing f (x) = cos(x + c)Summarize how adding c to the argument of a sinusoidal functionaﬀects the graph:1. f (x + c) → shifts the graph c units to the left2. f (x − c) → shifts the graph c units to the right31 of 45
92. 92. Graphing f (x) = cos(x) + dGiven: f (x) = cos xPlot the graph of f (x) = cos(x) + 1.32 of 45
93. 93. Graphing f (x) = cos(x) + dGiven: f (x) = cos xPlot the graph of f (x) = cos(x) + 1.32 of 45
94. 94. Graphing f (x) = cos(x) + dGiven: f (x) = cos xPlot the graph of f (x) = cos(x) + 1.32 of 45
95. 95. ExercisesDetermine the equation representing the graph below:Using the following functions:1. sine → f (x) = a · sin b(x + c) + d2. cosine → f (x) = a · cos b(x + c) + d33 of 45
96. 96. ExercisesAmplitude: 3Using the following functions:1. sine → f (x) = 3 · sin b(x + c) + d2. cosine → f (x) = 3 · cos b(x + c) + d33 of 45
97. 97. ExercisesPeriod: 2π → b = 1Using the following functions:1. sine → f (x) = 3 · sin 1(x + c) + d2. cosine → f (x) = 3 · cos 1(x + c) + d33 of 45
98. 98. ExercisesPhase shift (sine): π/4 to the rightUsing the following functions:1. sine → f (x) = 3 · sin(x − π/4) + d2. cosine → f (x) = 3 · cos(x + c) + d33 of 45
99. 99. ExercisesPhase shift (cosine): 3π/4 to the rightUsing the following functions:1. sine → f (x) = 3 · sin(x − π/4) + d2. cosine → f (x) = 3 · cos(x − 3π/4) + d33 of 45
100. 100. ExercisesVertical translation: 0Using the following functions:1. sine → f (x) = 3 · sin(x − π/4)2. cosine → f (x) = 3 · cos(x − 3π/4)33 of 45
101. 101. ExercisesDetermine the equation representing the graph below:Using the following functions:1. sine → f (x) = a · sin b(x + c) + d2. cosine → f (x) = a · cos b(x + c) + d34 of 45
102. 102. ExercisesVertical translation: −1Using the following functions:1. sine → f (x) = a · sin b(x + c)−12. cosine → f (x) = a · cos b(x + c)−134 of 45
103. 103. ExercisesAmplitude: 2Using the following functions:1. sine → f (x) = 2 · sin b(x + c) − 12. cosine → f (x) = 2 · cos b(x + c) − 134 of 45
104. 104. ExercisesPeriod: 2 → b = πUsing the following functions:1. sine → f (x) = 2 · sin π(x + c) − 12. cosine → f (x) = 2 · cos π(x + c) − 134 of 45
105. 105. ExercisesPhase shift (sine): 3/2 to the right, no phase shift for cosineUsing the following functions:1. sine → f (x) = 2 · sin π(x−3/2) − 12. cosine → f (x) = 2 · cos(πx) − 134 of 45
106. 106. Properties of the graph of y = tan(x):1. For what values of x is f (x) = tan(x) equal to zero?35 of 45
107. 107. Properties of the graph of y = tan(x):1. For what values of x is f (x) = tan(x) equal to zero? x = 0, π, 2π, 3π, ... or {x|x = nπ, n ∈ Z}35 of 45
108. 108. Properties of the graph of y = tan(x):1. For what values of x is f (x) = tan(x) equal to zero? x = 0, π, 2π, 3π, ... or {x|x = nπ, n ∈ Z}2. For what values of x is f (x) = tan(x) undeﬁned?35 of 45
109. 109. Properties of the graph of y = tan(x):1. For what values of x is f (x) = tan(x) equal to zero? x = 0, π, 2π, 3π, ... or {x|x = nπ, n ∈ Z}2. For what values of x is f (x) = tan(x) undeﬁned? x = π , 3π , 5π , ... or {x|x = 2 2 2 nπ 2 ,n is an odd integer }.35 of 45
110. 110. Properties of the graph of y = tan(x):Zeros: {x|x = nπ, n ∈ Z}Asymptotes:36 of 45
111. 111. Properties of the graph of y = tan(x):Zeros: {x|x = nπ, n ∈ Z}Asymptotes: {x|x = nπ , n is an odd integer }. 236 of 45
112. 112. Properties of the graph of y = tan(x):Zeros: {x|x = nπ, n ∈ Z}Asymptotes: {x|x = nπ , n is an odd integer }. 236 of 45
113. 113. Properties of the graph of y = tan(x):Domain:Range:Period:Increasing/Decreasing:37 of 45
114. 114. Properties of the graph of y = tan(x):Domain: {x|x = nπ , n is an odd integer }. 2Range:Period:Increasing/Decreasing:37 of 45
115. 115. Properties of the graph of y = tan(x):Domain: {x|x = nπ , n is an odd integer }. 2Range: {y|y ∈ R}.Period:Increasing/Decreasing:37 of 45
116. 116. Properties of the graph of y = tan(x):Domain: {x|x = nπ , n is an odd integer }. 2Range: {y|y ∈ R}.Period: πIncreasing/Decreasing:37 of 45
117. 117. Properties of the graph of y = tan(x):Domain: {x|x = nπ , n is an odd integer }. 2Range: {y|y ∈ R}.Period: πIncreasing/Decreasing: Increasing in all quadrants37 of 45
118. 118. Properties of the graph of y = cot(x):Recall: π cot x = tan −x 2 π = tan − x − 2 π = − tan x − 238 of 45
119. 119. Properties of the graph of y = cot(x): f (x) = tan(x)39 of 45
120. 120. Properties of the graph of y = cot(x): f (x) = tan(x − π/2)39 of 45
121. 121. Properties of the graph of y = cot(x): f (x) = − tan(x − π/2) = cot(x)39 of 45
122. 122. A comparison of y = tan(x) and y = cot(x) f (x) = tan(x) f (x) = cot(x)40 of 45
123. 123. Properties of the graph of y = cot(x):Domain:Range:Zeros:Period:Increasing/Decreasing:41 of 45
124. 124. Properties of the graph of y = cot(x):Domain: {x|x = nπ, n ∈ Z}Range:Zeros:Period:Increasing/Decreasing:41 of 45
125. 125. Properties of the graph of y = cot(x):Domain: {x|x = nπ, n ∈ Z}Range: {y|y ∈ R}.Zeros:Period:Increasing/Decreasing:41 of 45
126. 126. Properties of the graph of y = cot(x):Domain: {x|x = nπ, n ∈ Z}Range: {y|y ∈ R}.Zeros: {x|x = nπ , n is an odd integer } 2Period:Increasing/Decreasing:41 of 45
127. 127. Properties of the graph of y = cot(x):Domain: {x|x = nπ, n ∈ Z}Range: {y|y ∈ R}.Zeros: {x|x = nπ , n is an odd integer } 2Period: πIncreasing/Decreasing:41 of 45
128. 128. Properties of the graph of y = cot(x):Domain: {x|x = nπ, n ∈ Z}Range: {y|y ∈ R}.Zeros: {x|x = nπ , n is an odd integer } 2Period: πIncreasing/Decreasing: Decreasing in all quadrants41 of 45
129. 129. Graphing y = csc(x): f (x) = sin(x)42 of 45
130. 130. Graphing y = csc(x): f (x) = sin(x)42 of 45
131. 131. Graphing y = csc(x): f (x) = csc(x)42 of 45
132. 132. Graphing y = csc(x): f (x) = csc(x)42 of 45
133. 133. Properties of the graph of y = csc(x):Domain:Range:Zeros:Period:Increasing/Decreasing:43 of 45
134. 134. Properties of the graph of y = csc(x):Domain: {x|x = nπ, n ∈ Z}Range:Zeros:Period:Increasing/Decreasing:43 of 45
135. 135. Properties of the graph of y = csc(x):Domain: {x|x = nπ, n ∈ Z}Range: {y|y ≤ −1 ∪ y ≥ 1}.Zeros:Period:Increasing/Decreasing:43 of 45
136. 136. Properties of the graph of y = csc(x):Domain: {x|x = nπ, n ∈ Z}Range: {y|y ≤ −1 ∪ y ≥ 1}.Zeros: NonePeriod:Increasing/Decreasing:43 of 45
137. 137. Properties of the graph of y = csc(x):Domain: {x|x = nπ, n ∈ Z}Range: {y|y ≤ −1 ∪ y ≥ 1}.Zeros: NonePeriod: 2πIncreasing/Decreasing:43 of 45
138. 138. Properties of the graph of y = csc(x):Domain: {x|x = nπ, n ∈ Z}Range: {y|y ≤ −1 ∪ y ≥ 1}.Zeros: NonePeriod: 2πIncreasing/Decreasing: Increasing in Q2/Q3, Decreasing in Q1/Q443 of 45
139. 139. Graphing y = sec(x): f (x) = cos(x)44 of 45
140. 140. Graphing y = sec(x): f (x) = cos(x)44 of 45
141. 141. Graphing y = sec(x): f (x) = sec(x)44 of 45
142. 142. Graphing y = sec(x): f (x) = sec(x)44 of 45
143. 143. Properties of the graph of y = sec(x):Domain:Range:Zeros:Period:Increasing/Decreasing:45 of 45
144. 144. Properties of the graph of y = sec(x):Domain: {x|x = nπ , n is an odd integer } 2Range:Zeros:Period:Increasing/Decreasing:45 of 45
145. 145. Properties of the graph of y = sec(x):Domain: {x|x = nπ , n is an odd integer } 2Range: {y|y ≤ −1 ∪ y ≥ 1}.Zeros:Period:Increasing/Decreasing:45 of 45
146. 146. Properties of the graph of y = sec(x):Domain: {x|x = nπ , n is an odd integer } 2Range: {y|y ≤ −1 ∪ y ≥ 1}.Zeros: NonePeriod:Increasing/Decreasing:45 of 45
147. 147. Properties of the graph of y = sec(x):Domain: {x|x = nπ , n is an odd integer } 2Range: {y|y ≤ −1 ∪ y ≥ 1}.Zeros: NonePeriod: 2πIncreasing/Decreasing:45 of 45
148. 148. Properties of the graph of y = sec(x):Domain: {x|x = nπ , n is an odd integer } 2Range: {y|y ≤ −1 ∪ y ≥ 1}.Zeros: NonePeriod: 2πIncreasing/Decreasing: Increasing in Q1/Q2, Decreasing in Q3/Q445 of 45
149. 149. Any questions?46 of 45