Describing Graphs of Polynomial Functions

Before we begin, let's recall two important definitions.

• A function $f$ has even symmetry when $f(\text{-} x)=f(x)$ for all $x$ in its domain.
• A function $f$ has odd symmetry when $f(\text{-} x)=\text{-} f(x)$ for all $x$ in its domain.

Let's see how the graphs of these types of functions look. Consider the graphs above. We can see that for even symmetry, if $(x,y)$ is on the graph, then $(\text{-} x,y)$ is also on the graph. Meanwhile, for an odd symmetry, if $(x,y)$ is on the graph, then $(\text{-} x, \text{-} y)$ is also on the graph. Now, consider the given function. $\begin{gathered} s(x)=\text{-} x^3+2x-9 \end{gathered}$ Let's calculate $s({\color{#0000FF}{\text{-} x}}).$ Next, let's calculate ${\color{#FF0000}{\text{-}}} s(x).$ Finally, let's think about what these results tell us.

$s(x)$	$s(\text{-} x)$	$\text{-} s(x)$
$\text{-} x^3+2x-9$	$x^3-2x-9$	$x^3-2x+9$

Since $s(x)\neq s(\text{-} x)$ and $s(\text{-} x)\neq \text{-} s(x),$ the function $s$ has neither an even nor an odd symmetry.