Hyperbolas goes left-right when the y2
term is negative, and they go up-down when the x2
term is negative.
The asymptotes of an up-down hyperbola are y=bx/a,-bx/a, and y=ax/b,-ax/b for left-right.
Hyperbolas only have vertical and horizontal asymptotes when they are a reciprocal function, y=a/f(x).
Amazing what a 10 minute session of fiddling around with hyperbolas on WolframAlpha can teach you.
Really, if you need to know more, the best website that I know of is Wikipedia.