Sinusoidal regression worksheet

Assignment 7.1 Sinusoidal Function Models Total: 33 marks This is a hand-in assignment. Please clearly show your work and include units with final answers. If you are using technology to graph data and determine the regression equations, please include a printout of the graphs and equations or sketch and label the images neatly in your answers. Answers given without supporting calculations will not be awarded full marks. 1. The height above ground level, h, in metres, of a passenger on a Ferris wheel ride over time, t, in seconds, is modelled by the sinusoidal function π π h (t ) = 9.5 sin (t ) − + 13.5.  75 2 a) State the values for a, b, c, and d of the general form of the sinusoidal function. (2 marks) b) Determine the maximum and minimum height above ground attained by the rider, the rider’s variation from the average ride height during the ride, and the time it takes to complete one rotation on the Ferris wheel. (5 marks) M o dul e 7: Sinu s oidal Func t io n s 47 Assignment 7.1: Sinusoidal Function Models (continued) 2. Match the equations in the following set to their corresponding graphs. Explain the reasoning used to make your decisions. (10 marks) A   a) y = 3 sin ( 4 x )  b) y = 4 sin ( x ) + 3  c) y = sin ( π x ) − 4  d) y = −sin ( x )                      π  e) y = 3 sin  π ( x ) −   4     B C                  48          D   E                         Grade 12 Applied Mathematics Assignment 7.1: Sinusoidal Function Models (continued) a) y = 3 sin (4x) Graph Explanation: b) y = 4 sin (x) + 3 Graph Explanation: c) y = sin (px) - 4 Graph Explanation: d) y = -sin (x) Graph Explanation:   π  e) y = 3 sin  π ( x ) −   4   Graph Explanation: M o dul e 7: Sinu s oidal Func t io n s 49 Assignment 7.1: Sinusoidal Function Models (continued) 3. Norman and Lydia retired and moved to the Caribbean, where they enjoyed the beautiful sunrises each morning. The time sunrise occurred on six days during the year is recorded in the table. Aruba 1st Day of Month Jan. 1 Mar. 1 May 1 July 1 Oct. 1 Dec. 1 Time of Sunrise (a.m.) 7:02 6:45 6:21 6:14 6:44 7:03 Day of Year (#) 1 60 Time of Sunrise 7.033 6.75 a) Complete the table above by converting the date to the number of the day and the time to the decimal equivalent on the 24-hour clock. (2 marks) b) Use technology to plot the points and determine the sinusoidal regression equation that best models the time of sunrise each day over the year. Sketch the graph below or include a printout. Include labels and units on the graph. (4 marks) 50 Grade 12 Applied Mathematics Assignment 7.1: Sinusoidal Function Models (continued) c) What is the average time of sunrise during the year? (1 mark) d) Find the time of the earliest and latest sunrise during the year, and the dates on which these occur. How much do these times vary from the median time of sunrise during the year? (5 marks) e) What period is represented by this function? (2 marks) M o dul e 7: Sinu s oidal Func t io n s 51 Assignment 7.1: Sinusoidal Function Models (continued) f) Use the regression equation to approximate the time of sunrise on your birthday. Show your work. (2 marks) 52 Grade 12 Applied Mathematics ...