Graphing trigonometric functions

2,857 views

Published on

Slides to accompany lecture on graphing trigonometric functions.

Published in: Education, Technology
0 Comments
5 Likes
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
2,857
On SlideShare
0
From Embeds
0
Number of Embeds
90
Actions
Shares
0
Downloads
201
Comments
0
Likes
5
Embeds 0
No embeds

No notes for slide

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

×