Question

Are all \[y= \frac{1}{x^{2n-1}} \] hyperbolas?

Answer 1

I don't think so.
I believe the name hyperbola is given only to one of the conic sections.
It's equation can be written in one of these forms:\[\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\qquad\text{East-West Hyprbola}\]\[\frac{y^2}{a^2}-\frac{x^2}{b^2}=1\qquad\text{North-South Hyprbola}\]\[xy=m\qquad\text{Rectangular Hyperbola}\]

Answer 2

Thanks- it seemed as if conic-sections-wise http://en.wikipedia.org/wiki/File:Hyperbola_(PSF).png the higher powers corresponded to moving the slicer to the right.