Question

Write a cosine function that has a midline of 2, an amplitude of 3 and a period of 7π/4​.

PLS EXPLAIN! THXS!

Answer 1

`y=3\cos((8)/(7)x)+2`

Explanation:

Recall

• General cosine function:
  `y=a\cos(bx+c)+d`
• 
  `|a|` represents the amplitude, or the distance between the maximum/minimum of the function and the midline
• 
  `(2\pi)/(|b|)` represents the period, or the length of a cycle of the function which repeats continuously
• 
  `-(c)/(b)` is the phase shift/vertical shift
• 
  `d` is the midline, or the average between the maximum and the minimum, creating a horizontal center line

We are given that the amplitude is
`a=3` and that the equation of the midline is
`d=2`. Since we know the period to be
`(7\pi)/(4)`, we need to solve for
`b` by setting up the equation
`(7\pi)/(4)=(2\pi)/(|b|)`:

`(7\pi)/(4)=(2\pi)/(|b|)\\\\7\pi b=8\pi\\\\b=(8)/(7)`

Hence, our cosine function will be
`y=3\cos((8)/(7)x)+2`. See the attached graph for a visual.