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Sketching Polynomial Functions

Sketching Polynomial Functions 1.15 - Solution

We want to determine the end behavior of the graph of the given polynomial function. To do so, we will pay close attention to the leading term $ax^n,$ where $a$ is the leading coefficient and $n$ is the degree of the polynomial.Let's consider the given polynomial function. $\begin{gathered} f(x)=\text{-} 6x^4+2x^2 \end{gathered}$ Note that it is already written in standard form. We can see above that the leading coefficient is $\textcolor{darkorange}{\text{-} 6}$ and the degree is ${\color{#FF0000}{4}}.$ Let's now see how the leading coefficient and degree affect the end behavior of the graph of a polynomial function.
$\textcolor{darkorange}{a}>0,\$ ${\color{#FF0000}{n}}\ \text{even}$

$\textcolor{darkorange}{a}>0,\$ ${\color{#FF0000}{n}}\ \text{odd}$

$\textcolor{darkorange}{a}<0,\$ ${\color{#FF0000}{n}}\ \text{even}$

$\textcolor{darkorange}{a}<0,\$ ${\color{#FF0000}{n}}\ \text{odd}$

$\text{Reset}$

Since $\textcolor{darkorange}{\text{-}6 }<0$ and ${\color{#FF0000}{4}}$ is an even number, the end behavior of the given function is down and down. We can see above that as $x$ approaches negative infinity, $f(x)$ approaches negative infinity. As $x$ approaches positive infinity, $f(x)$ approaches negative infinity. $\begin{gathered} f(x) \rightarrow \text{-} \infty \text{ as } x \rightarrow \text{-} \infty \\ \textbf{and} \\ f(x) \rightarrow \text{-} \infty \text{ as } x \rightarrow + \infty \\ \end{gathered}$