Question

Write a cosine function that has an amplitude of 3, a midline of 5 and a period of 3/4. .

Answer 1

A cosine function is said to have an amplitude of 3, a midline of 5, and a period of 3/4.

It is required to write the cosine function.

Recall that the standard form of a cosine function is:

`y=a\cos(bx+c)+d`

Where a is the amplitude, 2π/b is the period, c is the horizontal shift, and d is the midline or vertical shift.

Equate the given period to 2π/b and solve for b:

`\begin{gathered} (2\pi)/(b)=(3)/(4) \\ \Rightarrow3b=8\pi \\ \Rightarrow(3b)/(3)=(8\pi)/(3) \\ \Rightarrow b=(8\pi)/(3) \end{gathered}`

Hence, substitute a=3, b=8π/3, and d=5 into the standard form of the cosine function:

`y=3\cos((8\pi)/(3)x+c)+5`

Since it is not given that the cosine function has a horizontal shift, substitute c=0 to get the required function:

`y=3\cos((8\pi)/(3)x)+5`

The function is y=3cos((8π/3) x)+5.