When looking at a quadratic function on a graph, whether it is down or up, the domain will always be all real numbers or does it depend on the vertex?

The domain of any graph depends on the vertex as well as the asymptotes. You can easily tell if the domain is all real numbers by looking at the graph and seeing that it goes clearly from - infinity to + infinity. If there is an asymptote that blocks it, then it will not be all real numbers.