Describe each of the following properties of the graph of the sine function, f(0) = sin(e),...

 

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Describe each of the following properties of the graph of the sine function, f(0) = sin(e), and relate the property to the unit circle definition of sine. Amplitude Period Domain Range x-intercepts Your response must include: A discussion of the amplitude, period, domain, and range of the sine or cosine function. A relation of each property to the unit circle. For example, if discussing the domain, describe how the domain of sine or cosine is based on the unit circle. . At least 100 words in complete sentences with appropriate grammar and spelling. Provide the source used for information (i.e. the website, the book, etc.). Not providing your source may result in an Academic Integrity violation. 1. Given the function defined by y = 3-6sin 2x + - 6 sin(2x+2). determine the solution to each question below. Show your work and explain how you determine each characteristic of the function. a. What is its period? b. What is the amplitude of its graph? c. What is its domain and range? d. What is the y-intercept of its graph? e. What is its phase shift? f. Sketch two periods of the graph of this function. Label the axes with key values. 2. The average high monthly temperature (in degrees Fahrenheit) in Duluth, Minnesota, is shown in the table. A scatterplot of the data is also given. a. 100+ 90- 80 70- 60 50 40 30+ 20 10- -10- Determine a function of the form f(x) = a sin b(x - c) + d, where a, b, c, and d are constants, that models the data. Show the work necessary in determining the values. Do not use regression capabilities of your graphing calculator. b. Determine the average high monthly temperature in Duluth Deg. F. Month 21 July 26 Aug 35 Sept 46 Oct 56 Nov 66 Dec 2 4 Month Jan Feb Mar Apr May June 6 8 10 Deg. F. 75 74 65 52 38 25 12 14 + 16 3. The rabbit population on Park Point in Duluth, MN at time t is modeled by r(t) 225cost +425 where t is measured in years. (Park Point, by the way, is reported as the world's longest freshwater sand spit) a. What is the maximum number of rabbits on Park Point during a population cycle? b. How long is the population cycle? c. Find the approximate number of rabbits on the island after 3.2 years. = d. Sketch a graph of the first period of this function, marking key axis values. e. Solve algebraically: in what year(s) during the first population cycle will the number of rabbits be equal to 500? 1. The number of miles s of roads cleared of snow in a 24-hour period in Duluth, MN is approximated by the 13 In model S = 25- with 2 h 28, where h is the depth of the snow in inches. In 3 a. Determine the number of miles of roads cleared of snow in 24 hours when h = 3 inches. b. If 17 miles of roads were cleared in 24 hours, how much snow fell in the recent snowstorm in Duluth? Describe each of the following properties of the graph of the sine function, f(0) = sin(e), and relate the property to the unit circle definition of sine. Amplitude Period Domain Range x-intercepts Your response must include: A discussion of the amplitude, period, domain, and range of the sine or cosine function. A relation of each property to the unit circle. For example, if discussing the domain, describe how the domain of sine or cosine is based on the unit circle. . At least 100 words in complete sentences with appropriate grammar and spelling. Provide the source used for information (i.e. the website, the book, etc.). Not providing your source may result in an Academic Integrity violation. 1. Given the function defined by y = 3-6sin 2x + - 6 sin(2x+2). determine the solution to each question below. Show your work and explain how you determine each characteristic of the function. a. What is its period? b. What is the amplitude of its graph? c. What is its domain and range? d. What is the y-intercept of its graph? e. What is its phase shift? f. Sketch two periods of the graph of this function. Label the axes with key values. 2. The average high monthly temperature (in degrees Fahrenheit) in Duluth, Minnesota, is shown in the table. A scatterplot of the data is also given. a. 100+ 90- 80 70- 60 50 40 30+ 20 10- -10- Determine a function of the form f(x) = a sin b(x - c) + d, where a, b, c, and d are constants, that models the data. Show the work necessary in determining the values. Do not use regression capabilities of your graphing calculator. b. Determine the average high monthly temperature in Duluth Deg. F. Month 21 July 26 Aug 35 Sept 46 Oct 56 Nov 66 Dec 2 4 Month Jan Feb Mar Apr May June 6 8 10 Deg. F. 75 74 65 52 38 25 12 14 + 16 3. The rabbit population on Park Point in Duluth, MN at time t is modeled by r(t) 225cost +425 where t is measured in years. (Park Point, by the way, is reported as the world's longest freshwater sand spit) a. What is the maximum number of rabbits on Park Point during a population cycle? b. How long is the population cycle? c. Find the approximate number of rabbits on the island after 3.2 years. = d. Sketch a graph of the first period of this function, marking key axis values. e. Solve algebraically: in what year(s) during the first population cycle will the number of rabbits be equal to 500? 1. The number of miles s of roads cleared of snow in a 24-hour period in Duluth, MN is approximated by the 13 In model S = 25- with 2 h 28, where h is the depth of the snow in inches. In 3 a. Determine the number of miles of roads cleared of snow in 24 hours when h = 3 inches. b. If 17 miles of roads were cleared in 24 hours, how much snow fell in the recent snowstorm in Duluth?