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# Describing Graphs of Polynomial Functions

## Describing Graphs of Polynomial Functions 1.11 - Solution

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.
Even functions

Odd functions

Reset

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 $s({\color{#0000FF}{\text{-} x}}).$
$s({\color{#0000FF}{\text{-} x}})=\text{-} ({\color{#0000FF}{\text{-} x}})^3+2({\color{#0000FF}{\text{-} x}})-9$
Simplify right-hand side
$s(\text{-} x)=\text{-} \left (\text{-} x^3\right)+2(\text{-} x)-9$
$s(\text{-} x)=x^3+2(\text{-} x)-9$
$s(\text{-} x)=x^3-2x-9$
Next, let's calculate ${\color{#FF0000}{\text{-}}} s(x).$
${\color{#FF0000}{\text{-}}} s(x)={\color{#FF0000}{\text{-}}} \left( \text{-} x^3+2x-9\right)$
$\text{-} s(x)=x^3-2x+9$
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.