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10.5 Write Trigonometric Functions and Models What is a sinusoid? How do you write a function for a sinusoid? How do you model a situation with a circular function? What is sinusoidal regression? Vocabulary Sinusoids are graphs of sine and cosine functions. Use the following equations and find the values of : (amplitude ) 2 the period ( > 0) to find b ℎ horizontal shift vertical shift = − ℎ + = cos − ℎ + Write a function for the sinusoid shown below. SOLUTION STEP 1 STEP 2 Find the maximum value M and minimum value m. From the graph, M = 5 and m = –1. Identify the vertical shift, k. The value of k is the mean of the maximum and minimum values. The vertical shift is M + m 5 + (–1) 4 k= = 2 = 2. = 2 2 So, k = 2. STEP 3 Decide whether the graph should be modeled by a sine or cosine function. Because the graph crosses the midline y = 2 on the y-axis, the graph is a sine curve with no horizontal shift. So, h = 0. STEP 4 Find the amplitude and period. The period is π = 2π So, b = 4. 2 b M – m 5 – (–1) 6 = 2 = 3. = The amplitude is a = 2 2 The graph is not a reflection, so a > 0. Therefore, a = 3. ANSWER The function is y = 3 sin 4x + 2. Jump Rope Circular Motion ROPE At a Double Dutch competition, two people swing jump ropes as shown in the diagram below. The highest point of the middle of each rope is 75 inches above the ground, and the lowest point is 3 inches. The rope makes 2 revolutions per second. Write a model for the height h (in feet) of a rope as a function of the time t (in seconds) if the rope is at its lowest point when t = 0. SOLUTION STEP 1 STEP 2 Find the maximum and minimum values of the function. A rope’s maximum height is 75 inches, so M = 75. A rope’s minimum height is 3 inches, so m = 3. Identify the vertical shift. The vertical shift for the model is: M + m 75 + 3 78 = = = = 39 k 2 2 2 STEP 3 Decide whether the height should be modeled by a sine or cosine function. When t = 0, the height is at its minimum. So, use a cosine function whose graph is a reflection in the x-axis with no horizontal shift (h = 0). STEP 4 Find the amplitude and period. ANSWER M – m 75 – 3 The amplitude is a = = 36. = 2 2 Because the graph is a reflection, a < 0. So, a = – 36. Because a rope is rotating at a rate of 2 revolutions per second, one revolution is completed in 0.5 second. So, the period is 2π = 0.5, and b = 4π. b A model for the height of a rope is h = – 36 cos 4π t + 39. Write a function for the sinusoid. 1. SOLUTION STEP 1 Find the maximum value M and minimum value m. From the graph, M = 2 and m = –2. STEP 2 Identify the vertical shift, k. The value of k is the mean of the maximum and minimum values. The vertical shift is M + m 2 + (–2) 0 k= = 2 = 0. = 2 2 So, k = 0. 1. STEP 3 Decide whether the graph should be modeled by a sine or cosine function. Because the graph peaks at y = 2 on the y-axis, the graph is a cos curve with no horizontal shift. So, h = 0. STEP 4 Find the amplitude and period. The period is 2π= 2π So, b = 3. 3 b M – m 2 – (–2) 4 = 2 = 2. = The amplitude is a = 2 2 The graph is not a reflection, so a > 0. Therefore, a = 2. The function is y = 2 cos 3x. ANSWER Write a function for the sinusoid. 2. SOLUTION STEP 1 Find the maximum value M and minimum value m. From the graph, M = 1 and m = –3. STEP 2 Identify the vertical shift, k. The value of k is the mean of the maximum and minimum values. The vertical shift is M + m 1 + (–3) 2 k= = 2 = 1. = 2 2 So, k = 1. 2. STEP 3 Decide whether the graph should be modeled by a sine or cosine function. Because the graph peaks at y = 1 when x = , the graph is a sin curve with a horizontal shift of 1. So, h = 1. STEP 4 Find the amplitude and period. The period is 2 So, b = π. M – m 1 – (–3) 4 = 2 = 2. = The amplitude is a = 2 2 The graph is not a reflection, so a > 0. Therefore, a = 2. ANSWER y = 2 sin π x – 1 Sinusoidal Regression Sinusoidal regression using a graphing calculator is another way to model sinusoids. The advantage is that the graphing calculator uses all of the data points to find the model. Energy The table below shows the number of kilowatt hours K (in thousands) used each month for a given year by a hangar at the Cape Canaveral Air Station in Florida. The time t is measured in months, with t = 1 representing January. Write a trigonometric model that gives K as a function of t. SOLUTION STEP 1 Enter the data in a graphing calculator. STEP 2 Make a scatter plot. STEP 3 Perform a sinusoidal regression, because the scatter plot appears sinusoidal. STEP 4 Graph the model and the data in the same viewing window. ANSWER The model appears to be a good fit. So, a model for the data is K = 23.9 sin (0.533t – 2.69) + 82.4. • What is a sinusoid? Graphs of sine and cosine functions • How do you write a function for a sinusoid? Find the values of a, b, h, and k for = − ℎ + and = cos − ℎ + • How do you model a situation with a circular function? Interpret the characteristics of the data to find the values of a, b, h, and k and determine whether the graph should be modeled by a sine function or cosine function. • What is sinusoidal regression? Another way to model sinusoids using a graphing calculator. 10.5 Assignment Page 648, 3-13 odd