Answer
Decreasing: $(0,\infty)$
Increasing: nowhere
Graph has no symmetry.
Work Step by Step
$\mathrm{First\:Part:}\:\:$ According to the definitions:
A function $\ f\ $ defined on an interval is increasing on $\ (a, b)\ $ if for every $\ x_1, x_2\ $ $\in$ $(a, b)$ $\ x_1\le x_2\ $ implies that $\ f(x_1)\le f(x_2).\ $
A function $\ f\ $ defined on an interval is decreasing on $\ (a, b)\ $ if for every $\ x_1, x_2\ $ $\in$ $(a, b)$ $\ x_1\le x_2\ $ implies that $\ f(x_1)\ge f(x_2).\ $
We can write the given function $\ y=-x^{\frac{3}{2}}\ $ as $\ y=-\sqrt{x^3}.$ The domain is $\ [0,\infty).$
First of all, create a table with a few points to sketch the graph.
$\quad \mathrm{See\:the\:table\:and\:graph\:above.}$
$\mathrm{Second\:Part:}\:\:$ Graph has no symmetry.
$\mathrm{Third\:Part:}\:\:$ The graph of the given function $\ y=-\sqrt{x^3}\ $ is decreasing on $\ (0,\infty).$
