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Explore properties, amplitudes, periods, frequencies, and more in sinusoidal functions. Learn to graph and interpret sinusoidal functions through real-world examples. Develop skills in horizontal and vertical dilations, reflections, translations, and modeling data.

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

  4. 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

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

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

  7. Let , where cos θ < 0. Find the exactvalues of the five remaining trigonometricfunctions of θ. A. B. C. D. 5–Minute Check 5

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

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

  10. sinusoid • amplitude • frequency • phase shift • vertical shift • midline Vocabulary

  11. Key Concept 1

  12. Key Concept 1

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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

  19. 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

  20. 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

  21. 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

  22. 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

  23. Key Concept 3

  24. 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 Graph Horizontal Dilations of Sinusoidal Functions Example 3

  25. 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 Graph Horizontal Dilations of Sinusoidal Functions Example 3

  26. 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

  27. 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

  28. Key Concept 4

  29. The period is the reciprocal of the frequency or . Use this value to find b. 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 0.75. 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| = 0.75. This means that a = ±0.75. Example 4

  30. Period formula period = Solve for |b|. |b| = 2π(50) or 100π Solve for b. Use Frequency to Write a Sinusoidal Function 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

  31. 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

  32. Key Concept 5

  33. 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 = . Frequency: Period: Graph Horizontal Translations of Sinusoidal Functions Amplitude: |a| = |2| or 2 Example 5

  34. 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. Graph Horizontal Translations of Sinusoidal Functions Example 5

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

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

  37. 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

  38. Period: Frequency: Phase shift: 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. Amplitude: |a| = | 1 | or 1 Vertical shift: dor 1 Midline: y = dor y = 1 Example 6

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

  40. 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

  41. Key Concept 7

  42. 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

  43. 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

  44. a = 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. The vertical shift d is the average of the maximum and minimum data values. Example 7

  45. 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 andxmin = day 10 Example 7

  46. Phase shift formula Phase shift = 17 and |b| = Solve for c. 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. Example 7

  47. 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: Modeling Data Using a Sinusoidal Function Example 7

  48. A. B. C. D. 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. Example 7

  49. End of the Lesson

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