Answer
The equation given is only symmetric about the $y$-axis
Work Step by Step
$y=x^{4}+x^{2}$
Test for symmetry about the $y$-axis by substituting $x$ by $-x$ and simplifying:
$y=(-x)^{4}+(-x)^{2}$
$y=x^{4}+x^{2}$
Since substituting $x$ by $-x$ yields an equivalent equation, the equation is symmetric about the $y$-axis.
Test for symmetry about the $x$-axis by substituting $y$ by $-y$ and simplifying:
$-y=x^{4}+x^{2}$
Since substituting $y$ by $-y$ does not yield an equivalent equation, the equation is not symmetric about the $x$-axis.
Test for symmetry about the origin by substituting $x$ by $-x$ and $y$ by $-y$ and simplifying:
$-y=(-x)^{4}+(-x)^{2}$
$-y=x^{4}+x^{2}$
Since substituting $x$ by $-x$ and $y$ by $-$ does not yield an equivalent equation, the equation is not symmetric about the origin.