Answer
The graph of the equation ${{y}^{5}}={{x}^{4}}+2$ is symmetric to the y-axis only.
Work Step by Step
Consider the equation, ${{y}^{5}}={{x}^{4}}+2$
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 ${{y}^{5}}={{x}^{4}}+2$.
$\begin{align}
& {{y}^{5}}={{\left( -x \right)}^{4}}+2 \\
& {{y}^{5}}={{x}^{4}}+2
\end{align}$
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 $y=-y$ in the equation ${{y}^{5}}={{x}^{4}}+2$.
$\begin{align}
& {{\left( -y \right)}^{5}}={{x}^{4}}+2 \\
& -{{y}^{5}}={{x}^{4}}+2
\end{align}$
Therefore, the equation is symmetric about the x-axis.
Now, check symmetry about the origin:
An equation is symmetric about the origin if $x=-x,y=-y$ and it leads to an equivalent equation, this implies that the function is symmetric about the origin.
Substitute $x=-x,y=-y$ in the equation ${{y}^{5}}={{x}^{4}}+2$.
$\begin{align}
& {{\left( -y \right)}^{5}}={{\left( -x \right)}^{4}}+2 \\
& -{{y}^{5}}={{x}^{4}}+2
\end{align}$
Therefore, the equation is not symmetric about the origin.
Hence, the equation is symmetric about the y-axis only.
