Parameterization of $\,y=\frac A{x^2}$ I am trying to parameterize the curve $\,y=\dfrac{A}{x^2}\,$ where $A$ is a constant and I have something that works but I'm not sure why. The parameterization of the curve is $\left(\sqrt[3]{\dfrac{A}{t}}\;,\,\sqrt[3]{t^2A}\right)$.
Can anyone help me understand why this works? I'm pretty lost at this point. I'm not even sure which branch of math would be best to field this question to but I've tagged a couple of different ones that I think deal with problems like this a lot.
Thanks in advance.

$y = \frac {A}{x^2}$ then  $x^2y = A$
So plugging $x = \sqrt [3]{\frac {A}{t}}, y=\sqrt [3]{A t^2}$
We indeed get that $x^2y = A$
But this is hardly the only parameterization that works.
$x = t, y = \frac {A}{t^2}$ is probably the most obvious parameterization.
or $x = \frac {1}{t}, y = At^2$ would also work.
