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Concept

# End Behavior

The end behavior of a function is the direction toward which $f(x)$ moves as $x$ extends to the left and the right infinitely. Analyzing each end individually, if $f(x)$ extends upward, the behavior is expressed as $f(x) \rightarrow + \infty.$ Alternatively, if $f(x)$ extends downward, the behavior is $f(x) \rightarrow \text{-} \infty.$ Consider the function $f(x).$ From the arrows on the graph, it can be seen that the left-end extends downward, while the right-end extends upward. Thus, the end behavior of $f$ can be expressed as follows. \begin{aligned} \text{As} \ x \rightarrow \text{-} \infty &\text{,} \ &&f(x) \rightarrow \text{-} \infty \\ \text{As} \ x \rightarrow + \infty &\text{,} \ &&f(x) \rightarrow + \infty \\ \end{aligned}

A polynomial function with a term with the highest degree of $ax^n,$ where $a$ is the leading coefficient and $n$ is the degree of the polynomial the end behavior depends on $a$ and $n.$ $\textcolor{blue}{a}>0,\$ ${\color{#FF0000}{n}}\ \text{even}$

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

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

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

$\text{Reset}$