What is the vertical displacement of the basic graph to produce a graph of
y = pi-3 cos(x - 2)?

There are an infinite number of them.
Here are a few:

2.1
2.2
2.3
2.3000000001
2.30000000000000006
2.5
2.00009
.
.
etc.

How to find them:

You probably never heard this before, but I'll bet
you'll remember it from now on:

ANY number that you can write down on paper completely, using digits
and a decimal point if you need it but no symbols, is a rational number.

If you can write it down completely without any 'comments', then it's rational.