I had typed up a response, but my browser crashed. Here:
I don't know if this is what you mean by "vertex," but usually the vertices of a curve are the points at which the derivative of its curvature is zero. The curvature of is
Differentiating,
So
and the vertices are at and
In general, the vertices of should be at and
Edit: I wasn't really thinking of this when I did the above, but the points along the axis of symmetry will be vertices, and is symmetric about the line So you could just find the points of intersection:
Much simpler!