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End Behavior

Concept

End Behavior

The end behavior of a function is the direction toward which f(x)f(x) moves as xx extends to the left and the right infinitely. Analyzing each end individually, if f(x)f(x) extends upward, the behavior is expressed as f(x)+. f(x) \rightarrow + \infty. Alternatively, if f(x)f(x) extends downward, the behavior is f(x)-. f(x) \rightarrow \text{-} \infty. Consider the function f(x).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 ff can be expressed as follows. As x-f(x)-As x+f(x)+\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 axn,ax^n, where aa is the leading coefficient and nn is the degree of the polynomial the end behavior depends on aa and n.n.

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

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

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

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

Reset\text{Reset}


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