Background
In a question I'm working on (course problem set for a Mathematics BSc, second year module in mechanics and fluids, unpublished), we're given a problem involving the surface
\begin{align} z & = \frac{1}{a^2}(4a^2-x^2-y^2)^\frac{3}{2} \\ & = \frac{1}{a^2}(4a^2-r^2)^\frac{3}{2} \end{align}
and, looking at the solution booklet, we're expected to recognise that this is symmetrical about the $z$-axis and use that fact in the solution.
Having put the formula into geogebra, I now know the shape is a downward-opening paraboloid*, and might recognise the formula in future. If I look at the formula, I can sort of see why it should be symmetric.
Edit: * Henry tells me this is wrong.
Question
Is there a way of expressing what it is about this formula that is sufficient to show symmetry? e.g. "$z$ is a polynomial in $r$, therefore the shape is symmetrical about the $z$-axis".
Request from comments:
If $z$ can be expressed as a function of $x^2$ and $y^2$ with no other $x$ or $y$ terms then there is reflection symmetry.
If $z$ can be expressed purely as a function of $r=\sqrt{x^2+y^2}$ with no $\theta$ or other $x$ or $y$ terms, then it is fully rotationally symmetry about the $z$-axis.
It may look like a paraboloid but thanks to the $\frac32$ exponent, it is not one