Answer
Graph is symmetric with respect to the origin.
The graph of the given function $\ y=-\frac{1}{x}\ $ is increasing on $\ (-\infty,0)\ $ and on $\ (0,\infty).$
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).\ $
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 is symmetric with respect to the origin.
$\mathrm{Third\:Part:}\:\:$ The graph of the given function $\ y=-\frac{1}{x}\ $ is increasing on $\ (-\infty,0)\ $ and on $\ (0,\infty).$
