Slides to accompany lecture on graphing trigonometric functions.

1. Graphing Trigonometric Functions
Mathematics 4
October 6, 2011
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2. Graphing the function y = sin x:
Identify the regions in the cartesian plane corresponding to the
quadrants of the unit circle:
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3. Graphing the function y = sin x:
Identify the regions in the cartesian plane corresponding to the
quadrants of the unit circle:
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4. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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5. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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6. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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7. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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8. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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9. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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10. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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11. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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12. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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13. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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14. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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15. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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16. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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17. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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18. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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19. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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20. Graphing the function y = sin x:
Plotting the sine values of the special angles:
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21. Expanding the graph of y = sin x:
Graphing beyond [0, 2π]:
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22. Properties of the graph of y = sin x:
Domain:
Range:
Zeros:
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23. Properties of the graph of y = sin x:
Domain: R
Range: y ∈ [−1, 1]
Zeros: {x|x = nπ, n ∈ Z}
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24. Properties of the graph of y = sin x:
Increasing in the following quadrants:
Decreasing in the following quadrants:
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25. Properties of the graph of y = sin x:
Increasing in the following quadrants: Q1 and Q4
Decreasing in the following quadrants: Q2 and Q3
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26. Properties of the graph of y = sin x:
Amplitude: One-half of the distance from the maximum to the
minimum value
The amplitude of y = sin x is:
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27. Properties of the graph of y = sin x:
Amplitude: One-half of the distance from the maximum to the
minimum value
The amplitude of y = sin x is: 1
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28. Properties of the graph of y = sin x:
Period: The distance from crest-to-crest or trough-to-trough
The period of y = sin x is:
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29. Properties of the graph of y = sin x:
Period: The distance from crest-to-crest or trough-to-trough
The period of y = sin x is 2π.
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30. Graphing y = sin(x − c):
Describe the graph of y = sin(x − π ).
2
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31. Graphing y = sin(x − c):
Describe the graph of y = sin(x − π ).
2
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32. Graphing y = sin(x − c):
Describe the graph of y = sin(x − π ).
2
π
The graph shifted 2 units to the right.
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33. Graphing y = cos x:
Recall:
π
cos x = sin −x
2
π
= sin − x −
2
π
= − sin x −
2
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34. Graphing y = cos x:
Graph of y = sin x
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35. Graphing y = cos x:
π
Graph of y = sin x − 2
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36. Graphing y = cos x:
π
Graph of y = − sin x − 2
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37. Graphing y = cos x:
π
Graph of y = − sin x − 2 = cos x
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38. Properties of y = cos x:
Domain:
Range:
Zeros:
Increasing in:
Decreasing in:
Amplitude:
Period:
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39. Properties of y = cos x:
Domain: R
Range:
Zeros:
Increasing in:
Decreasing in:
Amplitude:
Period:
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40. Properties of y = cos x:
Domain: R
Range: y ∈ [−1, 1]
Zeros:
Increasing in:
Decreasing in:
Amplitude:
Period:
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41. Properties of y = cos x:
Domain: R
Range: y ∈ [−1, 1]
Zeros: {x|x = nπ , n is an odd integer }
2
Increasing in:
Decreasing in:
Amplitude:
Period:
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42. Properties of y = cos x:
Domain: R
Range: y ∈ [−1, 1]
Zeros: {x|x = nπ , n is an odd integer }
2
Increasing in: Q3 and Q4
Decreasing in:
Amplitude:
Period:
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43. Properties of y = cos x:
Domain: R
Range: y ∈ [−1, 1]
Zeros: {x|x = nπ , n is an odd integer }
2
Increasing in: Q3 and Q4
Decreasing in: Q1 and Q2
Amplitude:
Period:
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44. Properties of y = cos x:
Domain: R
Range: y ∈ [−1, 1]
Zeros: {x|x = nπ , n is an odd integer }
2
Increasing in: Q3 and Q4
Decreasing in: Q1 and Q2
Amplitude: 1
Period:
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45. Properties of y = cos x:
Domain: R
Range: y ∈ [−1, 1]
Zeros: {x|x = nπ , n is an odd integer }
2
Increasing in: Q3 and Q4
Decreasing in: Q1 and Q2
Amplitude: 1
Period: 2π
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46. A comparison of y = sin x and y = cos x:
Identical properties:
Symmetry:
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47. A comparison of y = sin x and y = cos x:
Identical properties: Domain, Range, Amplitude, Period
Symmetry: y = sin x is symmetric wrt the origin. (Odd function)
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48. A comparison of y = sin x and y = cos x:
Identical properties: Domain, Range, Amplitude, Period
Symmetry: y = cos x is symmetric wrt the y-axis. (Even function)
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49. Pick-up quiz: 1 th sheet of paper.
4
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50. Pick-up quiz: 1 th sheet of paper.
4
1. What is the range of the sine and cosine functions?
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51. Pick-up quiz: 1 th sheet of paper.
4
1. What is the range of the sine and cosine functions?
2. Which function has its zeros at integer multiples of π?
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52. Pick-up quiz: 1 th sheet of paper.
4
1. 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:
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53. Pick-up quiz: 1 th sheet of paper.
4
1. 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?
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54. Pick-up quiz: 1 th sheet of paper.
4
1. 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?
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55. Graphing sinusoidal functions
The general form of a sinusoidal function is:
f (x) = a sin(b(x − c)) + d
or
f (x) = a cos(b(x − c)) + d
where a, b, c, and d modify the basic sine or cosine function.
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56. Graphing f (x) = a sin x
Given: f (x) = sin x
Plot the graph of f (x) = 2 sin x.
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57. Graphing f (x) = a sin x
Given: f (x) = sin x
Plot the graph of f (x) = 2 sin x.
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58. Graphing f (x) = a sin x
Given: f (x) = sin x
1
Plot the graph of f (x) = 2 sin x.
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59. Graphing f (x) = a sin x
Given: f (x) = sin x
1
Plot the graph of f (x) = 2 sin x.
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60. Graphing f (x) = a sin x
Given: f (x) = sin x
Plot the graph of f (x) = − 3 sin x.
2
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61. Graphing f (x) = a sin x
Given: f (x) = sin x
Plot the graph of f (x) = − 3 sin x.
2
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62. Graphing f (x) = a sin x
Summarize how multiplying a sinusoidal function by a affects the
graph:
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63. Graphing f (x) = a sin x
Summarize how multiplying a sinusoidal function by a affects the
graph:
1. |a| > 1 → expands graph vertically
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64. Graphing f (x) = a sin x
Summarize how multiplying a sinusoidal function by a affects the
graph:
1. |a| > 1 → expands graph vertically
2. |a| < 1 → compresses graph vertically
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65. Graphing f (x) = a sin x
Summarize how multiplying a sinusoidal function by a affects the
graph:
1. |a| > 1 → expands graph vertically
2. |a| < 1 → compresses graph vertically
3. a < 0 → flips the graph vertically
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66. Graphing f (x) = a sin x
Summarize how multiplying a sinusoidal function by a affects the
graph:
1. |a| > 1 → expands graph vertically
2. |a| < 1 → compresses graph vertically
3. a < 0 → flips the graph vertically
4. The amplitude of f (x) = a sin x is |a|
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67. Graphing f (x) = cos(b · x)
Given: f (x) = cos x
Plot the graph of f (x) = cos(2 · x).
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68. Graphing f (x) = cos(b · x)
Given: f (x) = cos x
Plot the graph of f (x) = cos(2 · x).
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69. Graphing f (x) = cos(b · x)
Given: f (x) = cos x
Plot the graph of f (x) = cos(2 · x).
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70. Graphing f (x) = cos(b · x)
Given: f (x) = cos x
Plot the graph of f (x) = cos( 1 · x).
2
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71. Graphing f (x) = cos(b · x)
Given: f (x) = cos x
Plot the graph of f (x) = cos( 1 · x).
2
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72. Graphing f (x) = cos(b · x)
Given: f (x) = cos x
Plot the graph of f (x) = cos( 1 · x).
2
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73. Graphing f (x) = sin(b · x)
Given: f (x) = sin x
Plot the graph of f (x) = sin(− 4 · x).
3
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74. Graphing f (x) = sin(b · x)
Given: f (x) = sin x
Plot the graph of f (x) = sin(− 4 · x).
3
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75. Graphing f (x) = sin(b · x)
Given: f (x) = sin x
Plot the graph of f (x) = sin(− 4 · x).
3
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76. Graphing f (x) = sin(b · x)
Given: f (x) = sin x
Plot the graph of f (x) = sin(− 4 · x).
3
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77. Graphing f (x) = sin(b · x)
Summarize how multiplying the argument of a sinusoidal function by
b affects the graph:
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78. Graphing f (x) = sin(b · x)
Summarize how multiplying the argument of a sinusoidal function by
b affects the graph:
1. |b| > 1 → compresses graph horizontally
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79. Graphing f (x) = sin(b · x)
Summarize how multiplying the argument of a sinusoidal function by
b affects the graph:
1. |b| > 1 → compresses graph horizontally
2. |b| < 1 → expands graph horizontally
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80. Graphing f (x) = sin(b · x)
Summarize how multiplying the argument of a sinusoidal function by
b affects the graph:
1. |b| > 1 → compresses graph horizontally
2. |b| < 1 → expands graph horizontally
3. b < 0 → flips the graph horizontally
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81. Graphing f (x) = sin(b · x)
Summarize how multiplying the argument of a sinusoidal function by
b affects the graph:
1. |b| > 1 → compresses graph horizontally
2. |b| < 1 → expands graph horizontally
3. b < 0 → flips the graph horizontally
2π
4. The period of f (x) = sin(b · x) is
|b|
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82. Graphing f (x) = a · sin(b · x)
Identify the amplitude and period, and sketch the graph:
2x
1. f (x) = cos 3
2. g(x) = 4 cos(2π · x)
3. h(x) = −2 sin(π · x)
4. f (x) = sin(−3x)
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83. Graphing f (x) = cos(x + c)
Given: f (x) = cos x
Plot the graph of f (x) = cos(x + π ).
3
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84. Graphing f (x) = cos(x + c)
Given: f (x) = cos x
Plot the graph of f (x) = cos(x + π ).
3
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85. Graphing f (x) = cos(x + c)
Given: f (x) = cos x
Plot the graph of f (x) = cos(x + π ).
3
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86. Graphing f (x) = cos(x + c)
Given: f (x) = cos x
5π
Plot the graph of f (x) = cos(x − 6 ).
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87. Graphing f (x) = cos(x + c)
Given: f (x) = cos x
5π
Plot the graph of f (x) = cos(x − 6 ).
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88. Graphing f (x) = cos(x + c)
Given: f (x) = cos x
5π
Plot the graph of f (x) = cos(x − 6 ).
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89. Graphing f (x) = cos(x + c)
Summarize how adding c to the argument of a sinusoidal function
affects the graph:
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90. Graphing f (x) = cos(x + c)
Summarize how adding c to the argument of a sinusoidal function
affects the graph:
1. f (x + c) → shifts the graph c units to the left
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91. Graphing f (x) = cos(x + c)
Summarize how adding c to the argument of a sinusoidal function
affects the graph:
1. f (x + c) → shifts the graph c units to the left
2. f (x − c) → shifts the graph c units to the right
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92. Graphing f (x) = cos(x) + d
Given: f (x) = cos x
Plot the graph of f (x) = cos(x) + 1.
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93. Graphing f (x) = cos(x) + d
Given: f (x) = cos x
Plot the graph of f (x) = cos(x) + 1.
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94. Graphing f (x) = cos(x) + d
Given: f (x) = cos x
Plot the graph of f (x) = cos(x) + 1.
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95. Exercises
Determine the equation representing the graph below:
Using the following functions:
1. sine → f (x) = a · sin b(x + c) + d
2. cosine → f (x) = a · cos b(x + c) + d
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96. Exercises
Amplitude: 3
Using the following functions:
1. sine → f (x) = 3 · sin b(x + c) + d
2. cosine → f (x) = 3 · cos b(x + c) + d
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97. Exercises
Period: 2π → b = 1
Using the following functions:
1. sine → f (x) = 3 · sin 1(x + c) + d
2. cosine → f (x) = 3 · cos 1(x + c) + d
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98. Exercises
Phase shift (sine): π/4 to the right
Using the following functions:
1. sine → f (x) = 3 · sin(x − π/4) + d
2. cosine → f (x) = 3 · cos(x + c) + d
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99. Exercises
Phase shift (cosine): 3π/4 to the right
Using the following functions:
1. sine → f (x) = 3 · sin(x − π/4) + d
2. cosine → f (x) = 3 · cos(x − 3π/4) + d
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100. Exercises
Vertical translation: 0
Using the following functions:
1. sine → f (x) = 3 · sin(x − π/4)
2. cosine → f (x) = 3 · cos(x − 3π/4)
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101. Exercises
Determine the equation representing the graph below:
Using the following functions:
1. sine → f (x) = a · sin b(x + c) + d
2. cosine → f (x) = a · cos b(x + c) + d
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102. Exercises
Vertical translation: −1
Using the following functions:
1. sine → f (x) = a · sin b(x + c)−1
2. cosine → f (x) = a · cos b(x + c)−1
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103. Exercises
Amplitude: 2
Using the following functions:
1. sine → f (x) = 2 · sin b(x + c) − 1
2. cosine → f (x) = 2 · cos b(x + c) − 1
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104. Exercises
Period: 2 → b = π
Using the following functions:
1. sine → f (x) = 2 · sin π(x + c) − 1
2. cosine → f (x) = 2 · cos π(x + c) − 1
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105. Exercises
Phase shift (sine): 3/2 to the right, no phase shift for cosine
Using the following functions:
1. sine → f (x) = 2 · sin π(x−3/2) − 1
2. cosine → f (x) = 2 · cos(πx) − 1
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106. Properties of the graph of y = tan(x):
1. For what values of x is f (x) = tan(x) equal to zero?
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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}
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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?
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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 }.
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110. Properties of the graph of y = tan(x):
Zeros: {x|x = nπ, n ∈ Z}
Asymptotes:
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111. Properties of the graph of y = tan(x):
Zeros: {x|x = nπ, n ∈ Z}
Asymptotes: {x|x = nπ , n is an odd integer }.
2
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112. Properties of the graph of y = tan(x):
Zeros: {x|x = nπ, n ∈ Z}
Asymptotes: {x|x = nπ , n is an odd integer }.
2
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113. Properties of the graph of y = tan(x):
Domain:
Range:
Period:
Increasing/Decreasing:
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114. Properties of the graph of y = tan(x):
Domain: {x|x = nπ , n is an odd integer }.
2
Range:
Period:
Increasing/Decreasing:
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115. Properties of the graph of y = tan(x):
Domain: {x|x = nπ , n is an odd integer }.
2
Range: {y|y ∈ R}.
Period:
Increasing/Decreasing:
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116. Properties of the graph of y = tan(x):
Domain: {x|x = nπ , n is an odd integer }.
2
Range: {y|y ∈ R}.
Period: π
Increasing/Decreasing:
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117. Properties of the graph of y = tan(x):
Domain: {x|x = nπ , n is an odd integer }.
2
Range: {y|y ∈ R}.
Period: π
Increasing/Decreasing: Increasing in all quadrants
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118. Properties of the graph of y = cot(x):
Recall:
π
cot x = tan −x
2
π
= tan − x −
2
π
= − tan x −
2
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119. Properties of the graph of y = cot(x):
f (x) = tan(x)
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120. Properties of the graph of y = cot(x):
f (x) = tan(x − π/2)
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121. Properties of the graph of y = cot(x):
f (x) = − tan(x − π/2) = cot(x)
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122. A comparison of y = tan(x) and y = cot(x)
f (x) = tan(x)
f (x) = cot(x)
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123. Properties of the graph of y = cot(x):
Domain:
Range:
Zeros:
Period:
Increasing/Decreasing:
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124. Properties of the graph of y = cot(x):
Domain: {x|x = nπ, n ∈ Z}
Range:
Zeros:
Period:
Increasing/Decreasing:
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125. Properties of the graph of y = cot(x):
Domain: {x|x = nπ, n ∈ Z}
Range: {y|y ∈ R}.
Zeros:
Period:
Increasing/Decreasing:
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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 }
2
Period:
Increasing/Decreasing:
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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 }
2
Period: π
Increasing/Decreasing:
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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 }
2
Period: π
Increasing/Decreasing: Decreasing in all quadrants
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129. Graphing y = csc(x):
f (x) = sin(x)
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130. Graphing y = csc(x):
f (x) = sin(x)
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131. Graphing y = csc(x):
f (x) = csc(x)
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132. Graphing y = csc(x):
f (x) = csc(x)
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133. Properties of the graph of y = csc(x):
Domain:
Range:
Zeros:
Period:
Increasing/Decreasing:
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134. Properties of the graph of y = csc(x):
Domain: {x|x = nπ, n ∈ Z}
Range:
Zeros:
Period:
Increasing/Decreasing:
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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:
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136. Properties of the graph of y = csc(x):
Domain: {x|x = nπ, n ∈ Z}
Range: {y|y ≤ −1 ∪ y ≥ 1}.
Zeros: None
Period:
Increasing/Decreasing:
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137. Properties of the graph of y = csc(x):
Domain: {x|x = nπ, n ∈ Z}
Range: {y|y ≤ −1 ∪ y ≥ 1}.
Zeros: None
Period: 2π
Increasing/Decreasing:
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138. Properties of the graph of y = csc(x):
Domain: {x|x = nπ, n ∈ Z}
Range: {y|y ≤ −1 ∪ y ≥ 1}.
Zeros: None
Period: 2π
Increasing/Decreasing: Increasing in Q2/Q3, Decreasing in Q1/Q4
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139. Graphing y = sec(x):
f (x) = cos(x)
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140. Graphing y = sec(x):
f (x) = cos(x)
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141. Graphing y = sec(x):
f (x) = sec(x)
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142. Graphing y = sec(x):
f (x) = sec(x)
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143. Properties of the graph of y = sec(x):
Domain:
Range:
Zeros:
Period:
Increasing/Decreasing:
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144. Properties of the graph of y = sec(x):
Domain: {x|x = nπ , n is an odd integer }
2
Range:
Zeros:
Period:
Increasing/Decreasing:
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145. Properties of the graph of y = sec(x):
Domain: {x|x = nπ , n is an odd integer }
2
Range: {y|y ≤ −1 ∪ y ≥ 1}.
Zeros:
Period:
Increasing/Decreasing:
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146. Properties of the graph of y = sec(x):
Domain: {x|x = nπ , n is an odd integer }
2
Range: {y|y ≤ −1 ∪ y ≥ 1}.
Zeros: None
Period:
Increasing/Decreasing:
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147. Properties of the graph of y = sec(x):
Domain: {x|x = nπ , n is an odd integer }
2
Range: {y|y ≤ −1 ∪ y ≥ 1}.
Zeros: None
Period: 2π
Increasing/Decreasing:
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148. Properties of the graph of y = sec(x):
Domain: {x|x = nπ , n is an odd integer }
2
Range: {y|y ≤ −1 ∪ y ≥ 1}.
Zeros: None
Period: 2π
Increasing/Decreasing: Increasing in Q1/Q2, Decreasing in Q3/Q4
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149. Any questions?
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