To determine the end behavior of the graph of the polynomial function [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex], we need to focus on the term with the highest degree since it will dominate the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex].
In this case, the term with the highest degree is [tex]\( 2x^3 \)[/tex]. We will analyze the behavior of [tex]\( f(x) \)[/tex] based on this term:
1. As [tex]\( x \rightarrow -\infty \)[/tex]:
[tex]\[
2x^3 \rightarrow 2(-\infty)^3 = 2(-\infty) = -\infty.
\][/tex]
Therefore, as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
2. As [tex]\( x \rightarrow \infty \)[/tex]:
[tex]\[
2x^3 \rightarrow 2(\infty)^3 = 2(\infty) = \infty.
\][/tex]
Therefore, as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
Based on this analysis, the end behavior of the graph of the polynomial function [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex] is as follows:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
So the correct choice is:
- As [tex]\( x \rightarrow -\infty, y \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
