Answer
The given equation has symmetry about the y-axis only.
Work Step by Step
Step I: To check symmetry about the y-axis:
Let us put $x=-x$ in the given equation; if the equation remains the same, then it has symmetry about the y-axis.
$\begin{align}
& y={{x}^{2}}+8 \\
& y={{\left( -x \right)}^{2}}+8 \\
& y={{x}^{2}}+8.
\end{align}$
It is the same as the provided equation, hence it has symmetry about the y-axis.
Step II: To check symmetry about the x-axis:
Let us put $y=-y$ in the given equation; if the equation remains the same, then it has symmetry about the x-axis.
$\begin{align}
& y={{x}^{2}}+8 \\
& \left( -y \right)={{x}^{2}}+8 \\
& -y={{x}^{2}}+8.
\end{align}$
It is not the same as the provided equation, hence it is not symmetric about the x-axis.
Step III: To check symmetry about the origin:
Let us put $x=-x\text{ and }y=-y$ in the given equation; if the equation remains the same, then it has symmetry about the origin.
$\begin{align}
& y={{x}^{2}}+8 \\
& \left( -y \right)={{\left( -x \right)}^{2}}+8 \\
& -y={{x}^{2}}+8
\end{align}$
It is not the same as the provided equation, hence it is not symmetric about the origin.
Therefore, the equation $y={{x}^{2}}+8$ has symmetry about the y-axis only.
