1. Splash Screen

2. Five-Minute Check (over Lesson 4-3) Then/Now New Vocabulary
Key Concept: Properties of the Sine and Cosine Functions
Key Concept: Amplitudes of Sine and Cosine Functions
Example 1: Graph Vertical Dilations of Sinusoidal Functions
Example 2: Graph Reflections of Sinusoidal Functions
Key Concept: Periods of Sine and Cosine Functions
Example 3: Graph Horizontal Dilations of Sinusoidal Functions
Key Concept: Frequency of Sine and Cosine Functions
Example 4: Real-World Example: Use Frequency to Write a Sinusoidal Function
Key Concept: Phase Shift of Sine and Cosine Functions
Example 5: Graph Horizontal Translations of Sinusoidal Functions
Example 6: Graph Vertical Translations of Sinusoidal Functions
Concept Summary: Graphs of Sinusoidal Functions
Example 7: Real-World Example: Modeling Data Using a Sinusoidal Function
Lesson Menu

3. Let (–5, 12) be a point on the terminal side of an angle θ in standard position. Find the exact values of the six trigonometric functions of θ. A. B. C. D.
5–Minute Check 1

4. Let (–5, 12) be a point on the terminal side of an angle θ in standard position. Find the exact values of the six trigonometric functions of θ. A. B. C. D.
5–Minute Check 1

5. Find the exact value of cot 3π, if defined
Find the exact value of cot 3π, if defined. If not defined, write undefined.
A. 0
B. 1
C. –1
D. undefined
5–Minute Check 2

6. Find the exact value of cot 3π, if defined
Find the exact value of cot 3π, if defined. If not defined, write undefined.
A. 0
B. 1
C. –1
D. undefined
5–Minute Check 2

7. Find the exact value of csc 210º.B. C.
D. 2
5–Minute Check 3

8. Find the exact value of csc 210º.B. C.
D. 2
5–Minute Check 3

9. Find the exact value of A. B. C. D.
5–Minute Check 4

10. Find the exact value of A. B. C. D.
5–Minute Check 4

11. Let , where cos θ < 0. Find the exact values of the five remaining trigonometric functions of θ.B. C.
5–Minute Check 5

12. Let , where cos θ < 0. Find the exact values of the five remaining trigonometric functions of θ.B. C.
5–Minute Check 5

13. Find the exact value of tan .B. C. D.
5–Minute Check 6

14. Find the exact value of tan .B. C. D.
5–Minute Check 6

15. You analyzed graphs of functions. (Lesson 1-5)
Graph transformations of the sine and cosine functions.
Use sinusoidal functions to solve problems.
Then/Now

16. sinusoid amplitude frequency phase shift vertical shift midline
Vocabulary

17. Key Concept 1

18. Key Concept 1

19. Graph Vertical Dilations of Sinusoidal Functions
Describe how the graphs of f (x) = sin x and g (x) = 2.5 sin x are related. Then find the amplitude of g (x), and sketch two periods of both functions on the same coordinate axes.
The graph of g (x) is the graph of f (x) expanded vertically. The amplitude of g (x) is |2.5| or 2.5.
Example 1

20. Graph Vertical Dilations of Sinusoidal Functions
Create a table listing the coordinates of the x-intercepts and extrema for f (x) = sin x for one period on [0, 2π]. Then use the amplitude of g (x) to find corresponding points on its graph.
Example 1

21. Graph Vertical Dilations of Sinusoidal Functions
Sketch the curve through the indicated points for each function. Then repeat the pattern suggested by one period of each graph to complete a second period on [2π, 4π]. Extend each curve to the left and right to indicate that the curve continues in both directions.
Example 1

22. Graph Vertical Dilations of Sinusoidal Functions
Answer:
Example 1

23. Graph Vertical Dilations of Sinusoidal Functions
Answer: The graph of g (x) is the graph of f (x) expanded vertically. The amplitude of g (x) is 2.5.
Example 1

24. A. The graph of g (x) is the graph of f (x) compressed horizontally.
Describe how the graphs of f (x) = cos x and g (x) = 5 cos x are related.
A. The graph of g (x) is the graph of f (x) compressed horizontally.
B. The graph of g (x) is the graph of f (x) compressed vertically.
C. The graph of g (x) is the graph of f (x) expanded horizontally.
D. The graph of g (x) is the graph of f (x) expanded vertically.
Example 1

25. A. The graph of g (x) is the graph of f (x) compressed horizontally.
Describe how the graphs of f (x) = cos x and g (x) = 5 cos x are related.
A. The graph of g (x) is the graph of f (x) compressed horizontally.
B. The graph of g (x) is the graph of f (x) compressed vertically.
C. The graph of g (x) is the graph of f (x) expanded horizontally.
D. The graph of g (x) is the graph of f (x) expanded vertically.
Example 1

26. Graph Reflections of Sinusoidal Functions
Describe how the graphs of f (x) = cos x and g (x) = –2 cos x are related. Then find the amplitude of g (x), and sketch two periods of both functions on the same coordinate axes.
The graph of g (x) is the graph of f (x) expanded vertically and then reflected in the x-axis. The amplitude of g (x) is |–2| or 2.
Example 2

27. Graph Reflections of Sinusoidal Functions
Create a table listing the coordinates of key points of f (x) = cos x for one period on [0, 2π]. Use the amplitude of g (x) to find corresponding points on the graph of y = 2 cos x. Then reflect these points in the x-axis to find corresponding points on the graph of g (x).
Example 2

28. Graph Reflections of Sinusoidal Functions
Sketch the curve through the indicated points for each function. Then repeat the pattern suggested by one period of each graph to complete a second period on [2π, 4π]. Extend each curve to the left and right to indicate that the curve continues in both directions.
Example 2

29. Graph Reflections of Sinusoidal Functions
Answer:
Example 2

30. Graph Reflections of Sinusoidal Functions
Answer: The graph of g (x) is the graph of f (x) expanded vertically and then reflected in the x-axis. The amplitude of g (x) is 2.
Example 2

31. Describe how the graphs of f (x) = cos x and g (x) = –6 cos x are related.
A. The graph of g (x) is the graph of f (x) expanded horizontally and then reflected in the y-axis.
B. The graph of g (x) is the graph of f (x) expanded vertically and then reflected in the x-axis.
C. The graph of g (x) is the graph of f (x) expanded horizontally and then reflected in the x-axis.
D. The graph of g (x) is the graph of f (x) expanded vertically and then reflected in the y-axis.
Example 2

32. Describe how the graphs of f (x) = cos x and g (x) = –6 cos x are related.
A. The graph of g (x) is the graph of f (x) expanded horizontally and then reflected in the y-axis.
B. The graph of g (x) is the graph of f (x) expanded vertically and then reflected in the x-axis.
C. The graph of g (x) is the graph of f (x) expanded horizontally and then reflected in the x-axis.
D. The graph of g (x) is the graph of f (x) expanded vertically and then reflected in the y-axis.
Example 2

33. Key Concept 3

34. Graph Horizontal Dilations of Sinusoidal Functions
Describe how the graphs of f (x) = cos x and g (x) = cos are related. Then find the period of g (x), and sketch at least one period of both functions on the same coordinate axes.
Because cos = cos , the graph of g (x) is the graph of f(x) expanded horizontally. The period of g (x) is
Example 3

35. Graph Horizontal Dilations of Sinusoidal Functions
Because the period of g (x) is 16π, to find corresponding points on the graph of g (x), change the x-coordinates of those key points on f (x) so that they range from 0 to 16π, increasing by increments of
Example 3

36. Graph Horizontal Dilations of Sinusoidal Functions
Sketch the curve through the indicated points for each function, continuing the patterns to complete one full cycle of each.
Example 3

37. A. The graph of g(x) is the graph of f(x) expanded vertically.
Describe how the graphs of f (x) = sin x and g (x) = sin 4x are related.
A. The graph of g(x) is the graph of f(x) expanded vertically.
B. The graph of g(x) is the graph of f(x) expanded horizontally.
C. The graph of g(x) is the graph of f(x) compressed vertically.
D. The graph of g(x) is the graph of f(x) compressed horizontally.
Example 3

38. A. The graph of g(x) is the graph of f(x) expanded vertically.
Describe how the graphs of f (x) = sin x and g (x) = sin 4x are related.
A. The graph of g(x) is the graph of f(x) expanded vertically.
B. The graph of g(x) is the graph of f(x) expanded horizontally.
C. The graph of g(x) is the graph of f(x) compressed vertically.
D. The graph of g(x) is the graph of f(x) compressed horizontally.
Example 3

39. Key Concept 4

40. Use Frequency to Write a Sinusoidal Function
MUSIC A bass tuba can hit a note with a frequency of 50 cycles per second (50 hertz) and an amplitude of Write an equation for a cosine function that can be used to model the initial behavior of the sound wave associated with the note.
The general form of the equation will be y = a cos bt, where t is the time in seconds. Because the amplitude is 0.75, |a| = This means that a = ±0.75.
The period is the reciprocal of the frequency or Use this value to find b.
Example 4

41. Period formula period = Solve for |b|. |b| = 2π(50) or 100π
Use Frequency to Write a Sinusoidal Function
Period formula
period =
Solve for |b|.
|b| = 2π(50) or 100π
Solve for b.
By arbitrarily choosing the positive values of a and b, one cosine function that models the initial behavior is y = 0.75 cos 100πt.
Answer:
Example 4

42. Answer: Sample answer: y = 0.75 cos 100πt
Use Frequency to Write a Sinusoidal Function
Period formula
period =
Solve for |b|.
|b| = 2π(50) or 100π
Solve for b.
By arbitrarily choosing the positive values of a and b, one cosine function that models the initial behavior is y = 0.75 cos 100πt.
Answer: Sample answer: y = 0.75 cos 100πt
Example 4

43. MUSIC In the equal tempered scale, F sharp has a frequency of 740 hertz. Write an equation for a sine function that can be used to model the initial behavior of the sound wave associated with F sharp having an amplitude of 0.2.
A. y = 0.2 sin 1480πt
B. y = 0.2 sin 740πt
C. y = 0.4 sin 370πt
D. y = 0.1 sin 74πt
Example 4

44. MUSIC In the equal tempered scale, F sharp has a frequency of 740 hertz. Write an equation for a sine function that can be used to model the initial behavior of the sound wave associated with F sharp having an amplitude of 0.2.
A. y = 0.2 sin 1480πt
B. y = 0.2 sin 740πt
C. y = 0.4 sin 370πt
D. y = 0.1 sin 74πt
Example 4

45. Key Concept 5

46. In this function, a = 2, b = 5, and c = .
Graph Horizontal Translations of Sinusoidal Functions
State the amplitude, period, frequency, and phase shift of Then graph two periods of the function.
In this function, a = 2, b = 5, and c = .
Amplitude: |a| = |2| or 2
Frequency:
Period:
Example 5

47. Graph Horizontal Translations of Sinusoidal Functions
Phase shift:
To graph , consider the graph of y = 2 sin 5x. The period of this function is Create a table listing the coordinates of key points of y = 2 sin 5x on the interval To account for a phase shift of , subtract from the x-values of each of the key points for the graph of y = 2 sin 5x.
Example 5

48. Graph Horizontal Translations of Sinusoidal Functions
Sketch the graph of y = 2 sin through these points, continuing the pattern to complete two cycles.
Example 5

49. Answer: Graph Horizontal Translations of Sinusoidal Functions
Example 5

50. Answer: amplitude = 2; period = ; frequency = ; phase shift =
Graph Horizontal Translations of Sinusoidal Functions
Answer: amplitude = 2; period = ; frequency = ; phase shift =
Example 5

51. State the amplitude, period, frequency, and phase shift of y = 4 cos
A. amplitude: 4, period: , frequency: ,phase shift:
B. amplitude: , period: 3, frequency: , phase shift:
C. amplitude: 4, period: 6π, frequency: , phase shift:
D. amplitude: –4, period: , frequency: , phase shift:
Example 5

52. State the amplitude, period, frequency, and phase shift of y = 4 cos
A. amplitude: 4, period: , frequency: ,phase shift:
B. amplitude: , period: 3, frequency: , phase shift:
C. amplitude: 4, period: 6π, frequency: , phase shift:
D. amplitude: –4, period: , frequency: , phase shift:
Example 5

53. In this function, a = 1, b = 1, c = π, and d = 1.
Graph Vertical Translations of Sinusoidal Functions
State the amplitude, period, frequency, phase shift, and vertical shift of y = sin (x + π) + 1. Then graph two periods of the function.
In this function, a = 1, b = 1, c = π, and d = 1.
Period:
Amplitude: |a| = | 1 | or 1
Frequency:
Phase shift:
Vertical shift: d or 1 Midline: y = d or y = 1
Example 6

54. Graph Vertical Translations of Sinusoidal Functions
Answer:
Example 6

55. Graph Vertical Translations of Sinusoidal Functions
Answer: amplitude = 1; period = 2π; frequency = ; phase shift = –π; vertical shift = 1
Example 6

56. State the amplitude, period, frequency, phase shift, and vertical shift of .
A. amplitude: 3, period: , frequency: , phase shift: , vertical shift: 2
B. amplitude: –3, period: , frequency: , phase shift: , vertical shift: –2
C. amplitude: 3, period: , frequency: , phase shift: , vertical shift: 2
D. amplitude: 3, period: , frequency: , phase shift: , vertical shift: –2
Example 6

57. State the amplitude, period, frequency, phase shift, and vertical shift of .
A. amplitude: 3, period: , frequency: , phase shift: , vertical shift: 2
B. amplitude: –3, period: , frequency: , phase shift: , vertical shift: –2
C. amplitude: 3, period: , frequency: , phase shift: , vertical shift: 2
D. amplitude: 3, period: , frequency: , phase shift: , vertical shift: –2
Example 6

58. Key Concept 7

59. Modeling Data Using a Sinusoidal Function
METEOROLOGY The tides in the Bay of Fundy, in New Brunswick, Canada, have extreme highs and lows everyday. The table shows the high tides for one lunar month. Write a trigonometric function that models the height of the tides as a function of time x, where x = 1 represents the first day of the month.
Example 7

60. Step 1 Make a scatter plot of the data and choose a model.
Modeling Data Using a Sinusoidal Function
Step 1 Make a scatter plot of the data and choose a model.
The graph appears wave-like, so you can use a sinusoidal function of the form y = a sin (bx + c) + d or y = a cos (bx + c) + d to model the data. We will choose to use y = a cos (bx + c) + d to model the data.
Example 7

61. Modeling Data Using a Sinusoidal Function
Step 2 Find the maximum M and minimum m values of the data, and use these values to find a, b, c, and d.
The maximum and minimum heights are 28.0 and 23.3, respectively. The amplitude a is half of the distance between the extrema.
a =
The vertical shift d is the average of the maximum and minimum data values.
Example 7

62. Because the period equals , you can write |b| = Therefore, | b | =
Modeling Data Using a Sinusoidal Function
A sinusoid completes half of a period in the time it takes to go from its maximum to its minimum value. One period is twice this time.
Period = 2(xmax – xmin) = 2(17 – 10) or 14 xmax = day 17 and xmin = day 10
Because the period equals , you can write |b| = Therefore, | b | =
Example 7

63. Modeling Data Using a Sinusoidal Function
The maximum data value occurs when x = 17. Since y = cos x attains its first maximum when x = 0, we must apply a phase shift of 17 – 0 or 17 units. Use this value to find c.
Phase shift formula
Phase shift = 17 and |b| =
Solve for c.
Example 7

64. y = 2.35 cos is one model for the height of the tides.
Modeling Data Using a Sinusoidal Function
Step 3 Write the function using the values for a, b, c, and d. Use b = .
y = 2.35 cos is one model for the height of the tides.
Answer:
Example 7

65. y = 2.35 cos is one model for the height of the tides.
Modeling Data Using a Sinusoidal Function
Step 3 Write the function using the values for a, b, c, and d. Use b = .
y = 2.35 cos is one model for the height of the tides.
Answer:
Example 7

66. TEMPERATURES The table shows the average monthly high temperatures for Chicago. Write a function that models the high temperatures using x = 1 to represent January. A. B. C. D.
Example 7

67. TEMPERATURES The table shows the average monthly high temperatures for Chicago. Write a function that models the high temperatures using x = 1 to represent January. A. B. C. D.
Example 7

68. End of the Lesson