Answer
There is symmetry only about the x-axis.
See graph.
The intercepts: $(-5,0)$, $(0,-\sqrt5)$, $(0,\sqrt5)$.
Work Step by Step
$$x=y^2-5$$
Testing for the symmetry about the x-axis:
$$x=(-y)^2-5$$ $$x=y^2-5$$
Since the resulting equation is the same as the original equation, there is symmetry about the x-axis.
Testing for the symmetry about the y-axis:
$$-x=y^2-5$$ $$x=-y^2+5$$
Since the resulting equation is not the same as the original equation, there is no symmetry about the y-axis.
Testing for the symmetry about the origin:
$$-x=(-y)^2-5$$ $$-x=y^2-5$$ $$x=-y^2+5$$
Since the resulting equation is not the same as the original equation, there is no symmetry about the origin.
For $y=0$:
$$x=0^2-5=-5$$
For $x=0$:
$$0=y^2-5$$ $$y^2=5$$ $$y_1=-\sqrt5$$ $$y_2=\sqrt5$$
Thus, three points on the curve are at $(-5,0)$, $(0,-\sqrt5)$ and $(0,\sqrt5)$.
Using the three points, the graph is as shown.
The intercepts are as for x-intercept is $(-5,0)$, and for y-intercepts are $(0,-\sqrt5)$ and $(0,\sqrt5)$.
