Answer
The intercept with $y$ axis is $10$. There is no intercept with $x$ axis. The graph is symmetric. The plot in on the figure.
Work Step by Step
To get the intercept with $y$ axis we put $x=0$ and solve for $y$:
$$y=\frac{10}{0^2 +1} = \frac{10}{1}= 10.$$
To get the intercept with $x$ axis we put $y=0$ and solve for $x$:
$$0=\frac{10}{x^2+1}.$$
Multiplying both sides by $x^2+1$ we get
$$10=0$$ which is clearly not possible so there is no solution for $x$ i.e. no intercept with $x$ axis.
To test for symmetry we put $-x$ instead of $x$:
$$y(-x) = \frac{10}{(-x)^2+1} = \frac{10}{x^2+1} = y(x)$$
where we used $(-x)^2 = x^2$ for any number $x$. Since $y(x) = y(-x)$ the graph is symmetric with respect to the $y$ axis.
