From simple questions to complex issues, IDNLearn.com has the answers you need. Join our platform to receive prompt and accurate responses from experienced professionals in various fields.

What is the end behavior of the graph of the polynomial function [tex]f(x) = 2x^3 - 26x - 24[/tex]?

A. As [tex]x \rightarrow -\infty, y \rightarrow -\infty[/tex] and as [tex]x \rightarrow \infty, y \rightarrow -\infty[/tex]

B. As [tex]x \rightarrow -\infty, y \rightarrow -\infty[/tex] and as [tex]x \rightarrow \infty, y \rightarrow \infty[/tex]

C. As [tex]x \rightarrow -\infty, y \rightarrow \infty[/tex] and as [tex]x \rightarrow \infty, y \rightarrow -\infty[/tex]

D. As [tex]x \rightarrow -\infty, y \rightarrow \infty[/tex] and as [tex]x \rightarrow \infty, y \rightarrow \infty[/tex]


Sagot :

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].