for instance, what is the vertical shift for:

Question

y = 1/2secx + 1

*** by the way, that is secx + 1, not sec(x + 1)

Answer 1

y = f(x) + 1

would move the graph of y = f(x) up by 1, no matter what the graph is.

So, yes, the secant graph would indeed move up 1 unit.