Question

Over the past several years, the owner of a boutique on Aspen Avenue has observed a pattern in the amount of revenue for the store. The revenue reaches a maximum of about $ 50000 in June and a minimum of about $ 29000 in December. Suppose the months are numbered 1 through 12, and write a function of the form f(x)=Asin(B[x−C])+D that models the boutique's revenue during the year, where x corresponds to the month. If needed, you can enter π=3.1416... as 'pi' in your answer. Please show your work/explain your steps!

Answer 1

f(x) = 10500sin(π/6(x -3)) +39500

Explanation:

The average of the maximum and minimum revenue is the vertical offset of the function, parameter D.

D = (50,000 +29,000)/2 = 39,500

The amplitude of the function is the difference between the maximum and the offset.

A = 50,000 -D = 50,000 -39,500 = 10,500

The horizontal scale factor B is a number that will be equal to 2π when x-C = 12:

12B = 2π

B = π/6 . . . . . . divide by 12

The horizontal offset is such that revenue is neutral and increasing at the value x=C. That will be in the month of March, when x=3, so C=3.

Now we have all the parameters, so we can write the equation:

f(x) = 10500sin(π/6(x -3)) +39500