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 $ 42000 in april and a minimum of about $ 25000 in october. 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 corresponds to the month.

The sinusoidal function that models the revenue, in thousands, is given by:

The standard form of the function is:

In which:

In this problem, first, we consider that the revenue is between $25,000 and $42000, thus, the amplitude is of $17,000, then:

In a standard equation, this means that the function would assume values between -8.5 and 8.5. We want them to be between 25 and 42, thus, a vertical shift of is used.

We want a period of 12 months, which is 1 year, thus:

Using a graphing calculator, the maximum revenue is for and the minumum revenue is for . We want it for and , so a shift to the right of 1 unit, thus the horizontal shift is

Then, the equation is:

Which graph is sketched at the end of this answer.

A similar problem is given at link

Step-by-step explanation:

Given information:

Maximum revenue = 42000

Minimum revenue = $25000

Time period = 12 months

We need to write a function that models the boutique's revenue during the year, where corresponds to the month.

          .... (1)

where, A is amplitude, is period, C is phase shift and D is midline.

Substitute the value of A, B and D in equation (1).

        ..... (2)

In April, revenue of the store is $42000. So, the graph passes through the point (4,42000).

Divide both sides by 8500.

On comparing both sides we get

Substitute the value of C in equation (2).

Therefore, the required function is .

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below:

$9.41

step-by-step explanation: