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Describing Graphs of Polynomial Functions

Describing Graphs of Polynomial Functions 1.11 - Solution

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

  • A function ff has even symmetry when f(-x)=f(x)f(\text{-} x)=f(x) for all xx in its domain.
  • A function ff has odd symmetry when f(-x)=-f(x)f(\text{-} x)=\text{-} f(x) for all xx in its domain.
Let's see how the graphs of these types of functions look.
Even functions

Odd functions


Consider the graphs above. We can see that for even symmetry, if (x,y)(x,y) is on the graph, then (-x,y)(\text{-} x,y) is also on the graph. Meanwhile, for an odd symmetry, if (x,y)(x,y) is on the graph, then (-x,-y)(\text{-} x, \text{-} y) is also on the graph. Now, consider the given function. s(x)=-x3+2x9\begin{gathered} s(x)=\text{-} x^3+2x-9 \end{gathered} Let's calculate s(-x).s({\color{#0000FF}{\text{-} x}}).
s(-x)=-(-x)3+2(-x)9s({\color{#0000FF}{\text{-} x}})=\text{-} ({\color{#0000FF}{\text{-} x}})^3+2({\color{#0000FF}{\text{-} x}})-9
Simplify right-hand side
s(-x)=-(-x3)+2(-x)9s(\text{-} x)=\text{-} \left (\text{-} x^3\right)+2(\text{-} x)-9
s(-x)=x3+2(-x)9s(\text{-} x)=x^3+2(\text{-} x)-9
s(-x)=x32x9s(\text{-} x)=x^3-2x-9
Next, let's calculate -s(x).{\color{#FF0000}{\text{-}}} s(x).
-s(x)=-(-x3+2x9){\color{#FF0000}{\text{-}}} s(x)={\color{#FF0000}{\text{-}}} \left( \text{-} x^3+2x-9\right)
-s(x)=x32x+9\text{-} s(x)=x^3-2x+9
Finally, let's think about what these results tell us.
s(x)s(x) s(-x)s(\text{-} x) -s(x)\text{-} s(x)
-x3+2x9\text{-} x^3+2x-9 x32x9x^3-2x-9 x32x+9x^3-2x+9

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