## Calculus: Early Transcendentals (2nd Edition)

The function is only symmetric about the $y$-axis
$f(x)=x^{4}+5x^{2}-12$ $\textbf{Symmetry about the$x$-axis}$ This function has no symmetry about the $x$-axis because if it had, it would violate the Vertical Rule Test. Another way to realize this is that changing $f(x)$ by $-f(x)$ does not yield an equivalent function. $\textbf{Symmetry about the$y$-axis}$ Substitute $x$ by $-x$ in $f(x)$ and simplify: $f(-x)=(-x)^{4}+5(-x)^{2}-12=...$ $...=x^{4}+5x^{2}-12$ Since $f(-x)=f(x)$, this function is symmetric about the $y$-axis. $\textbf{Symmetry about the origin}$ This function has no symmetry about the origin beacuse, as it was seen when testing for $y$-axis symmetry, $f(-x)$ is not equal to $-f(x)$. The graph of this function is: