Answer
The graph of the equation ${{y}^{2}}={{x}^{2}}+6$ symmetric to the x-axis, y-axis and origin.
Work Step by Step
Consider the equation, ${{y}^{2}}={{x}^{2}}+6$
Check symmetry about the y-axis:
An equation is symmetric about the y-axis if $-x$ is substituted in the function and the result is an equivalent equation then the graph of the equation is symmetric with respect to the y-axis
Substitute $x=-x$ in the equation,
$\begin{align}
& {{\left( -y \right)}^{2}}={{x}^{2}}+6 \\
& {{y}^{2}}={{x}^{2}}+6
\end{align}$
Therefore, the equation is symmetric about the y-axis
Now, check symmetry about the x-axis:
An equation is symmetric about the x-axis, if $-y$ is substituted in the function and it leads to an equivalent equation than the graph is symmetric with respect to the x-axis.
Substitute $y=-y$ ,
$\begin{align}
& {{\left( -y \right)}^{2}}={{x}^{2}}+6 \\
& {{y}^{2}}={{x}^{2}}+6
\end{align}$
Therefore, the equation is symmetric about the x-axis.
Now, check symmetry about the origin:
An equation is symmetric about origin if $x=-x,y=-y$ and it leads to an equivalent equation, this implies that the function is symmetric about the origin.
Substitute $x=-x,y=-y$
$\begin{align}
& {{\left( -y \right)}^{2}}={{\left( -x \right)}^{2}}+6 \\
& {{y}^{2}}={{x}^{2}}+6
\end{align}$
Therefore, the equation is not symmetric about the origin
Hence, the equation is symmetric about the x-axis, y-axis and origin.
