To determine the end behavior of the polynomial function [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex], we analyze the dominating term of the polynomial as [tex]\( x \)[/tex] becomes very large positively ([tex]\( x \rightarrow \infty \)[/tex]) and very large negatively ([tex]\( x \rightarrow -\infty \)[/tex]).
### Analyzing as [tex]\( x \to \infty \)[/tex]:
1. The leading term of the polynomial is [tex]\( 2x^3 \)[/tex].
2. As [tex]\( x \to \infty \)[/tex], the term [tex]\( 2x^3 \)[/tex] dominates all other terms.
3. Since [tex]\( 2x^3 \)[/tex] grows very large and positive as [tex]\( x \to \infty \)[/tex] (because the coefficient 2 is positive), [tex]\( f(x) \to \infty \)[/tex] as [tex]\( x \to \infty \)[/tex].
### Analyzing as [tex]\( x \to -\infty \)[/tex]:
1. Again, consider the leading term [tex]\( 2x^3 \)[/tex].
2. As [tex]\( x \to -\infty \)[/tex], [tex]\( (-x)^3 = -x^3 \)[/tex] becomes very large and negative.
3. So [tex]\( 2(-x^3) = -2x^3 \)[/tex] also grows very large and negative as [tex]\( x \to -\infty \)[/tex], which means [tex]\( f(x) \to -\infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
Given these analyses, we conclude:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex].
Therefore, the correct description for the end behavior of the polynomial [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex] is:
- As [tex]\( x \rightarrow -\infty, y \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
Thus, the answer aligns with the second given option:
- As [tex]\( x \rightarrow -\infty, y \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
