Question

What is the period and amplitude for `f(x) = 2cos(4x + pi) -1`?

Answer 1

You have the form:
`y=`Amplitude`*cos((2pi)/(period)x+....)`
So in your case:
Amplitude=`2`
Period=`(2pi)/4=pi/2`

`+pi` is an initial phase and `-1` is a vertical shift.

Graphically:
graph{2cos(4x+pi)-1 [-10, 10, -5, 5]}

Note that your `cos` is displaced downwards and now oscillates around `y=-1`! It also starts at `-1` as `cos(0+pi)`.