Discover new information and get your questions answered with IDNLearn.com. Ask any question and get a detailed, reliable answer from our community of experts.
Sagot :
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].
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].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thank you for choosing IDNLearn.com. We’re committed to providing accurate answers, so visit us again soon.