Answer
The graph of the equation ${{x}^{3}}-{{y}^{2}}=5$ symmetric to the x-axis only.
Work Step by Step
Consider the equation,
$\begin{align}
& {{x}^{3}}-{{y}^{2}}=5 \\
& {{x}^{2}}={{y}^{2}}+5 \\
\end{align}$
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,
$\begin{align}
& {{x}^{3}}={{y}^{2}}+5 \\
& {{\left( -x \right)}^{3}}={{y}^{2}}+5 \\
& -{{x}^{3}}={{y}^{2}}+5
\end{align}$
Therefore, the equation is not 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 $y=-y$ ,
$\begin{align}
& {{x}^{3}}={{\left( -y \right)}^{2}}+5 \\
& ={{y}^{2}}+5
\end{align}$
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 $x=-x,y=-y$
$\begin{align}
& {{\left( -x \right)}^{3}}={{\left( -y \right)}^{2}}+5 \\
& -{{x}^{3}}={{y}^{2}}+5
\end{align}$
Therefore, the equation is not symmetric about the origin
Hence, the equation is symmetric about the x-axis.
