Describing the equivalence classes of (x,y)R(u,v) <=> y-v=x^2-u^2

Question

The equivalence classes of the relation

$(x,y)R(u,v) \leftrightarrow y - v = x^2 - u^2$

is supposed to look like parables with minimums on the y-axis. Can anyone see why this is?

Answer 1

For every $c$, consider the parabola $K(c)$ of equation $y=x^2+c$. Each set $K(c)$ is a parabola symmetric with respect to the $y$-axis and with minimum at $(0,c)$. For every $(x,y)$, the equivalence class of $(x,y)$ is $K(y-x^2)$.