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# 0504 ch 5 day 4

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### 0504 ch 5 day 4

1. 1. 5.3 Trigonometric GraphsMatthew 6:33 But seek ﬁrst his kingdom and hisrighteousness, and all these things will be added unto you.
2. 2. y = sin x
3. 3. y = sin x } } } } Q1 Q2 Q3 Q4 Unit Circle
4. 4. y = sin xMax: 1Min: -1 } } } } Q1 Q2 Q3 Q4 Unit Circle
5. 5. y = sin xMax: 1Min: -1 max− minamplitude = =1 2 } } } } Q1 Q2 Q3 Q4 Unit Circle
6. 6. y = sin xMax: 1Min: -1 max− minamplitude = =1 2 } } } }1 cycle occurs in 2π Q1 Q2 Q3 Q4∴ Period : 2π Unit Circle
7. 7. y = sin xMax: 1Min: -1 max− minamplitude = =1 2 } } } }1 cycle occurs in 2π Q1 Q2 Q3 Q4∴ Period : 2π Unit CircleDomain : {x : x ∈R}Range : {y : −1 ≤ y ≤ 1}
8. 8. y = sin xMax: 1Min: -1 max− minamplitude = =1 2 } } } }1 cycle occurs in 2π Q1 Q2 Q3 Q4∴ Period : 2π Unit CircleDomain : {x : x ∈R} Use the 5 key pointsRange : {y : −1 ≤ y ≤ 1} to help you graph
9. 9. y = cos x
10. 10. y = cos x } } } } Q1 Q2 Q3 Q4 Unit Circle Use the 5 key points to help you graph
11. 11. y = cos xMax: 1Min: -1amplitude: 1Period: 2π } } } } Q1 Q2 Q3 Q4 Unit Circle Use the 5 key points to help you graph
12. 12. y = cos xMax: 1Min: -1amplitude: 1Period: 2π } } } }Domain : {x : x ∈R} Q1 Q2 Q3 Q4Range : {y : −1 ≤ y ≤ 1} Unit Circle Use the 5 key points to help you graph
13. 13. y = cos x Max: 1 Min: -1 amplitude: 1 Period: 2π } } } } Domain : {x : x ∈R} Q1 Q2 Q3 Q4 Range : {y : −1 ≤ y ≤ 1} Unit Circle πIf you shift cosθ right , Use the 5 key points 2it looks just like sin θ . to help you graph πThey are out of phase by . 2
14. 14. y = cos x Max: 1 Min: -1 amplitude: 1 Period: 2π } } } } Domain : {x : x ∈R} Q1 Q2 Q3 Q4 Range : {y : −1 ≤ y ≤ 1} Unit Circle πIf you shift cosθ right , Use the 5 key points 2it looks just like sin θ . to help you graph πThey are out of phase by . 2 ⎛ π ⎞ sin θ = cos ⎜ θ − ⎟ ⎝ 2 ⎠
15. 15. Sinusoidal Functions
16. 16. Sinusoidal Functions y = asinb ( x − c ) + d
17. 17. Sinusoidal Functions y = asinb ( x − c ) + d a is the amplitude if a < 0 , the graph is reﬂected about the x-axis
18. 18. Sinusoidal Functions y = asinb ( x − c ) + d a is the amplitude if a < 0 , the graph is reﬂected about the x-axis b is related to the period in this way: normal period period = b
19. 19. Sinusoidal Functions y = asinb ( x − c ) + d a is the amplitude if a < 0 , the graph is reﬂected about the x-axis b is related to the period in this way: normal period period = b c is the horizontal shift (or ‘phase shift’) if c < 0 , shifted left if c > 0 , shifted right
20. 20. Sinusoidal Functions y = asinb ( x − c ) + d a is the amplitude if a < 0 , the graph is reﬂected about the x-axis b is related to the period in this way: normal period period = b c is the horizontal shift (or ‘phase shift’) if c < 0 , shifted left if c > 0 , shifted right d is the vertical shift
21. 21. Discuss and Graph1. y = 3cosθ
22. 22. Discuss and Graph1. y = 3cosθ amp : 3
23. 23. Discuss and Graph1. y = 3cosθ amp : 3 per : 2π
24. 24. Discuss and Graph1. y = 3cosθ amp : 3 per : 2π H .S. : none
25. 25. Discuss and Graph1. y = 3cosθ amp : 3 per : 2π H .S. : none V.S. : none
26. 26. Discuss and Graph1. y = 3cosθ amp : 3 per : 2π H .S. : none V.S. : noneKnow how to graph on trig graph paper using the5 key points. Verify with your calculator.
27. 27. Discuss and Graph ⎛ π ⎞2. y = −2sin ⎜ x + ⎟ ⎝ 2 ⎠
28. 28. Discuss and Graph ⎛ π ⎞2. y = −2sin ⎜ x + ⎟ ⎝ 2 ⎠ amp : 2
29. 29. Discuss and Graph ⎛ π ⎞2. y = −2sin ⎜ x + ⎟ ⎝ 2 ⎠ amp : 2 per : 2π
30. 30. Discuss and Graph ⎛ π ⎞2. y = −2sin ⎜ x + ⎟ ⎝ 2 ⎠ amp : 2 per : 2π π H .S. : left 2
31. 31. Discuss and Graph ⎛ π ⎞2. y = −2sin ⎜ x + ⎟ ⎝ 2 ⎠ amp : 2 per : 2π π H .S. : left 2 V.S. : none
32. 32. Discuss and Graph ⎛ π ⎞2. y = −2sin ⎜ x + ⎟ ⎝ 2 ⎠ amp : 2 per : 2π π H .S. : left 2 V.S. : none Reﬂected about the x-axis
33. 33. Discuss and Graph3. y = sin ( 2x − π )
34. 34. Discuss and Graph3. y = sin ( 2x − π ) Factor out the 2
35. 35. Discuss and Graph3. y = sin ( 2x − π ) Factor out the 2 ⎛ π ⎞ y = sin 2 ⎜ x − ⎟ ⎝ 2 ⎠
36. 36. Discuss and Graph3. y = sin ( 2x − π ) Factor out the 2 ⎛ π ⎞ y = sin 2 ⎜ x − ⎟ ⎝ 2 ⎠ amp : 1
37. 37. Discuss and Graph3. y = sin ( 2x − π ) Factor out the 2 ⎛ π ⎞ y = sin 2 ⎜ x − ⎟ ⎝ 2 ⎠ amp : 1 per : π
38. 38. Discuss and Graph3. y = sin ( 2x − π ) Factor out the 2 ⎛ π ⎞ y = sin 2 ⎜ x − ⎟ ⎝ 2 ⎠ amp : 1 norm. per. per : π period = b 2π p= =π 2
39. 39. Discuss and Graph3. y = sin ( 2x − π ) Factor out the 2 ⎛ π ⎞ y = sin 2 ⎜ x − ⎟ ⎝ 2 ⎠ amp : 1 norm. per. per : π period = b π 2π H .S. : right p= =π 2 2
40. 40. Discuss and Graph3. y = sin ( 2x − π ) Factor out the 2 ⎛ π ⎞ y = sin 2 ⎜ x − ⎟ ⎝ 2 ⎠ amp : 1 norm. per. per : π period = b π 2π H .S. : right p= =π 2 2 V.S. : none
41. 41. Discuss and Graph 1 ⎛ 1 ⎞4. y = cos ⎜ x + π ⎟ − 1 2 ⎝ 2 ⎠
42. 42. Discuss and Graph 1 ⎛ 1 ⎞4. y = cos ⎜ x + π ⎟ − 1 2 ⎝ 2 ⎠ 1 1 y = cos ( x + 2π ) − 1 2 2
43. 43. Discuss and Graph 1 ⎛ 1 ⎞4. y = cos ⎜ x + π ⎟ − 1 2 ⎝ 2 ⎠ 1 1 y = cos ( x + 2π ) − 1 2 2 1 amp : 2
44. 44. Discuss and Graph 1 ⎛ 1 ⎞4. y = cos ⎜ x + π ⎟ − 1 2 ⎝ 2 ⎠ 1 1 y = cos ( x + 2π ) − 1 2 2 1 amp : 2 per : 4π
45. 45. Discuss and Graph 1 ⎛ 1 ⎞4. y = cos ⎜ x + π ⎟ − 1 2 ⎝ 2 ⎠ 1 1 y = cos ( x + 2π ) − 1 2 2 1 amp : norm. per. 2 period = b per : 4π 2π p= = 4π 1 2
46. 46. Discuss and Graph 1 ⎛ 1 ⎞4. y = cos ⎜ x + π ⎟ − 1 2 ⎝ 2 ⎠ 1 1 y = cos ( x + 2π ) − 1 2 2 1 amp : norm. per. 2 period = b per : 4π 2π p= = 4π H .S. : 2π left 1 2
47. 47. Discuss and Graph 1 ⎛ 1 ⎞4. y = cos ⎜ x + π ⎟ − 1 2 ⎝ 2 ⎠ 1 1 y = cos ( x + 2π ) − 1 2 2 1 amp : norm. per. 2 period = b per : 4π 2π p= = 4π H .S. : 2π left 1 2 V.S. : 1 down
48. 48. Discuss and Graph 1 ⎛ 1 ⎞4. y = cos ⎜ x + π ⎟ − 1 2 ⎝ 2 ⎠ 1 1 y = cos ( x + 2π ) − 1 2 2 1 amp : norm. per. 2 period = b per : 4π 2π p= = 4π H .S. : 2π left 1 2 V.S. : 1 down Use the 5 key points to help you graph this!
49. 49. HW #4Unless you’re willing to have a go, fail miserably,and have another go, success won’t happen. Phillip Adams