SECTION 6.6 Phase Shift; Sinusoidal Curve Fitting 457 SUMMARY Steps for Fitting a Sine Function y A x B sin ω φ ( ) = − + to Data Step 1 Determine A, the amplitude of the function. = − Amplitude largest datavalue smallestdata value 2 Step 2 Determine B, the vertical shift of the function. = + Vertical shift largest data value smallest data value 2 Step 3 Determine ω. Since the period T, the time it takes for the data to repeat, is π ω = T 2 , we have ω π = T 2 Step 4 Determine the horizontal shift of the function by using the period of the data. Divide the period into four subintervals of equal length. Determine the x-coordinate for the maximum of the sine function and the x-coordinate for the maximum value of the data. Use this information to determine the value of the phase shift, φ ω . Now Work PROBLEM 29(a)–(c) Let’s look at another example. Since the number of hours of sunlight in a day cycles annually, the number of hours of sunlight in a day for a given location can be modeled by a sinusoidal function. The longest day of the year (in terms of hours of sunlight) occurs on the day of the summer solstice. For locations in the Northern Hemisphere, the summer solstice is the time when the Sun is farthest north. In 2022, the summer solstice occurred on June 21 (the 172nd day of the year) at 5:14 a.m. EDT. The shortest day of the year occurs on the day of the winter solstice, the time when the Sun is farthest south (for locations in the Northern Hemisphere). In 2022, the winter solstice occurred on December 21 (the 355th day of the year) at 4:48 p.m. EST. Finding a Sinusoidal Function for Hours of Daylight According to the Old Farmer’s Almanac, the number of hours of sunlight in Boston on the day of the summer solstice is 15.28, and the number of hours of sunlight on the day of the winter solstice is 9.07. (a) Find a sinusoidal function of the form ω φ ( ) = − + y A x B sin that fits the data. (b) Use the function found in part (a) to predict the number of hours of sunlight in Boston on April 1, the 91st day of the year. (c) Use a graphing utility to graph the function found in part (a). (d) Look up the number of hours of sunlight for April 1 in the Old Farmer’s Almanac and compare it to the results found in part (b). Source:The Old Farmer’s Almanac, www.almanac.com EXAMPLE 4 (continued)
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