Answer
The Amplitude is $1$
The Period is $ 0.5$
Work Step by Step
For
$y=a\sin c(x−h) + k$ $y =a \cos c(x−h) + k$,
Amplitude: $|a|$, Period: $\frac{2\pi} {|c|}$,
Given $y = -2+ \cos (4\pi x)$
The Amplitude is $|1| = 1$
The Period is $\frac{2\pi}{|4\pi|} = 0.5$
The transformations are as follows
The blue graph represents the graph of $\cos x$
The red graph represents the graph of the given function that compressed horizontally by a factor of $4 \pi$ and shifted down 2 units
See graph below.