Let W be the fraction of the width of the moon illuminated (as we observe it) as a function of time / in days. For example, when the moon is full, W = 1. One model is W = =+ = 1/2 + 1/cos (255) 29.5 This model assumes that t = 0 corresponds to a full moon. This model for W represents a sine wave that has been shifted vertically. State the period of the sine wave. (Give an exact answer. Use symbolic notation and fractions where needed.) period: Explain in terms of phases of the moon what this period means. This period represents the number of days between the phase of the new moon and the phase of the full moon. This period represents the number of days it takes the moon to get from any phase to that same phase again. This period represents the number of days between the phase of the new moon and the phase of the half moon. Use the model for W to determine how much of the width of the moon is illuminated 2 days after a full moon. (Give your answer to the nearest whole number.) W = The following observation is taken from the website of the Astronomical Applications Department of the U.S. Naval Observatory: days % Although Full Moon occurs each month at a specific date and time, the Moon's disk may appear to be full for several nights in a row if it is clear. This is because the percentage of the Moon's disk that appears illuminated changes very slowly around the time of Full Moon (also around New Moon, but the Moon is not visible at all then). Explain this observation in terms of a graph of the function W and rates of change. Use the graphing utility to plot the graph of W.

Describe how the graph of the transformed sine function : f(t) = -3 sin(2t) + 1 can be obtained from the "parent" sine wave. Then graph both functions on the graph below; label which is which.

Solve

1. One cycle of a sine function begins at x = 3 and ends at x = a. Determine the period of the function. b. Determine the phase shift of the function. c. Write the equation of the function in the form y = sin k(x - d)