## Precalculus (6th Edition) Blitzer

The graph of the equation $x={{y}^{2}}+6$ is symmetric to the x-axis only.
Consider the equation, $x={{y}^{2}}+6$ 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. Substitute $x=-x$ in the equation, \begin{align} & -x={{y}^{2}}+6 \\ & -x={{y}^{2}}+6 \end{align} Therefore, the equation is not symmetric about the y-axis. Now, check symmetry about the x-axis: A equation is symmetric about the x-axis, if $-y$ is substituted in the function and it leads to an equivalent equation. Substitute $y=-y$ , \begin{align} & x={{\left( -y \right)}^{2}}+6 \\ & ={{y}^{2}}+6 \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. Substitute $x=-x,y=-y$ \begin{align} & -x={{\left( -y \right)}^{2}}+6 \\ & -x={{y}^{2}}+6 \\ \end{align} Therefore, the equation is not symmetric about the origin. Hence, the equation is only symmetric about the x-axis.