How would you represent $y=(x-h)^2+k$ in polar coordinates? I tried using $$x=r\cos(\theta)$$ and $$y=r\sin(\theta)$$ and ended up with $r\sin(\theta) = (r\cos(\theta)-h)^2 + k$ and wasn't sure how to proceed from there.

Set $$x-h=\sqrt{k} \tan(\theta)$$
So,
$$x=h+\sqrt{k} \tan(\theta)$$
Now, $$y=k \tan^2(\theta) + k = k\,{\sec^2(\theta)} $$
Finally,
$x=h+\sqrt{k} \tan(\theta)$
and
$y=k\,{\sec^2(\theta)} $.
