Consider y = sec(x). Now form g(x) = sec(3x) + 2. Does adding +2 change the locations of vertical asymptotes?

Yes. The +2 shifts zeros of the denominator upward and thus moves the asymptotes to new x-values where cos(3x) = -1/2.

Yes. A vertical translation will remove asymptotes if the shift is large enough to keep the function finite everywhere.

No. Vertical asymptotes depend on where cosine of the argument equals zero (cos(3x)=0); adding a vertical shift +2 moves the graph up but does not affect where the denominator is zero.

No; however, it changes the period to 2π/(3) because the +2 alters horizontal scaling.