The first function, `y=2^x` has no special type of symmetry. It's neither odd nor even, which can be verified algebraically but is easiest seen by graphing:
The only asymptote is the line `y=0,` since as `x->-oo,` `2^x` approaches zero. As `x->+oo,` the graph just keeps rising and rising, so there's no asymptote in the positive direction.
The second function is even, so it is its own mirror image when reflected about the `y` axis, as seen in the graph:
This also follows algebraically, since `|-x|=|x|.` Now, whether it has asymptotes depends on how you define "asymptote", so you'll have to check your book or your notes for that. If "asymptote" means a line that the graph gets arbitrarily close to but never touches (which is a bad definition in my opinion, but it is what the word means when translated and Apollonius, who first used the term, meant it in this way), then there are no asymptotes.
If "asymptote" means a line whose distance to the graph approaches zero as `x` approaches plus or minus infinity (see the link), then the graph has two asymptotes, `y=x` and `y=-x.` This is clearly true because the distance between these lines and the graph of `y=|x|` is always zero (if you're on the right side of the `y` axis), so the distance "approaches" zero trivially.