write a cosecant function with a period of 4pi, phase shift of 4 pi, and vertical shift of 8.

My answer:csc(theta/2-8pi)-8

first of all, a function is an equation, you have no equal sign

start with y = csc(Ø/2)
now a phase shift of 4π
y = csc (1/2)(Ø - 4π)
now a vertical shift of 8
y = csc (1/2)(Ø-4Ø) + 8 or y = csc (Ø/2 - 2π) + 8

To write a cosecant function with a period of 4π, phase shift of 4π, and vertical shift of 8, we need to start with the general form of the cosecant function and then apply the changes.

The general form of the cosecant function is given as:

csc(theta)

The period of a trigonometric function is the length it takes for one complete cycle. In this case, the period is 4π, so we need to adjust the argument of the cosecant function to stretch the cycle to cover that period. To accomplish this, we divide the angle theta by 2, resulting in:

csc(theta/2)

Next, to introduce the phase shift of 4π, we need to adjust the argument further. We subtract 4π from the argument to shift the graph to the right by 4π, giving us:

csc((theta/2) - 4π)

Finally, to account for the vertical shift of 8 units, we subtract 8 from the whole function, resulting in:

csc((theta/2) - 4π) - 8

Therefore, the cosine function with a period of 4π, phase shift of 4π, and vertical shift of 8 is:

csc((theta/2) - 4π) - 8