Test the graph of $y=\dfrac{4x^{2}}{x^{2}+1}$ for symmetry.

Solution  $x$-axis: To test for symmetry with respect to the $x$-axis, we replace $y$ with $-y$. Since $-y=\dfrac{4x^{2}}{x^{2}+1}$ is not equivalent to $y=\dfrac{4x^{2}}{x^{2}+1}$, the graph of the equation is not symmetric with respect to the $x$-axis.

$y$-axis: To test for symmetry with respect to the $y$-axis, we replace $x$ with $-x$. Since $y=\dfrac{4( -x) ^{2}}{( -x) ^{2}+1}= \dfrac{4x^{2}}{x^{2}+1}$ is equivalent to $y=\dfrac{4x^{2}}{x^{2}+1}$, the graph of the equation is symmetric with respect to the $y$-axis.

Origin: To test for symmetry with respect to the origin, we replace $x$ with $-x$ and $y$ with $-y$. $$\begin{eqnarray*} -y &=&\dfrac{4( -x) ^{2}}{( -x) ^{2}+1}\quad{\color{#0066A7}{\hbox{Replace x by -x and y by -y.}}} \\[1pt] -y &=&\dfrac{4x^{2}}{x^{2}+1}\quad {\color{#0066A7}{\hbox{Simplify.}}} \\[1pt] y &=&-\dfrac{4x^{2}}{x^{2}+1}\quad{\color{#0066A7}{\hbox{Multiply both sides by -1.}}} \end{eqnarray*}$$