First of all, infinity is a concept, not a number. It is the idea of numbers going on without ever ending. (Source: my maths teacher from last year, who had a Ph. D, so I'll assume she's very credible.)
Meanwhile, negative infinity is basically the idea of numbers going LOWER infinitely. It's used a lot as shorthand in notation of the domain or range of an equation, stating that the domain or range continues to become lower without end.
Finally, that first graph you used as "proof" is essentially just x^3=y when x=/=0, or something like that. My maths class is starting on this sort of thing in just a couple of months, so Ballom is probably the best source here.
~ BioGaia