To determine the end behavior of the polynomial function [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex], we need to analyze how the function behaves as [tex]\( x \)[/tex] approaches positive and negative infinity.
In polynomials, the term with the highest degree (which has the largest exponent) dominates the behavior of the function as [tex]\( x \)[/tex] becomes very large (positively or negatively). In this case, the term with the highest degree in the polynomial [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex] is [tex]\( 2x^3 \)[/tex].
1. As [tex]\( x \to -\infty \)[/tex]:
- The leading term [tex]\( 2x^3 \)[/tex] will dominate the behavior of [tex]\( f(x) \)[/tex].
- When [tex]\( x \)[/tex] is a large negative number, [tex]\( x^3 \)[/tex] will be a large negative number as well.
- Multiplying [tex]\( 2x^3 \)[/tex] by a positive coefficient 2 and a negative cube of [tex]\( x \)[/tex], we get [tex]\( 2x^3 \to -\infty \)[/tex].
Thus, as [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex].
2. As [tex]\( x \to \infty \)[/tex]:
- Similarly, for large positive values of [tex]\( x \)[/tex], the term [tex]\( 2x^3 \)[/tex] will again be the dominant term.
- When [tex]\( x \)[/tex] is a large positive number, [tex]\( x^3 \)[/tex] will be a large positive number.
- Multiplying [tex]\( 2x^3 \)[/tex] by a positive coefficient 2, we get [tex]\( 2x^3 \to \infty \)[/tex].
Thus, as [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
Putting these together, the end behavior of the polynomial function [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex] is:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
Therefore, the correct description of the end behavior is:
- As [tex]\( x \rightarrow-\infty, y \rightarrow-\infty \)[/tex].
- As [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
