Testing for Symmetry

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*}