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To determine the end behavior of the polynomial function [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex], we'll analyze how the function behaves as [tex]\( x \)[/tex] approaches [tex]\(-\infty\)[/tex] and [tex]\( \infty \)[/tex]. Specifically, we look at the leading term of the polynomial, which dominates the behavior of the function for large absolute values of [tex]\( x \)[/tex].
### Step-by-Step Analysis:
1. Identify the leading term:
The leading term in the polynomial [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex] is the term with the highest degree, which is [tex]\( 2x^3 \)[/tex].
2. Analyze the leading term for end behavior:
- The leading term [tex]\( 2x^3 \)[/tex] will have the most significant impact on the value of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(\pm\infty\)[/tex].
- Since it is a cubic term with a positive coefficient, it affects the end behavior as follows:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( 2x^3 \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( 2x^3 \rightarrow \infty \)[/tex].
3. Combine the effects of the leading term with the entire polynomial:
- The other terms in the polynomial, [tex]\( -26x \)[/tex] and [tex]\( -24 \)[/tex], become relatively insignificant compared to [tex]\( 2x^3 \)[/tex] for large absolute values of [tex]\( x \)[/tex].
4. Conclude the end behavior:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
Based on this analysis, the correct description of the end behavior of the polynomial function [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex] is:
- As [tex]\( x \rightarrow -\infty, y \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
So, the correct end behavior is:
As [tex]\( x \rightarrow -\infty, y \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
### Step-by-Step Analysis:
1. Identify the leading term:
The leading term in the polynomial [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex] is the term with the highest degree, which is [tex]\( 2x^3 \)[/tex].
2. Analyze the leading term for end behavior:
- The leading term [tex]\( 2x^3 \)[/tex] will have the most significant impact on the value of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\(\pm\infty\)[/tex].
- Since it is a cubic term with a positive coefficient, it affects the end behavior as follows:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( 2x^3 \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( 2x^3 \rightarrow \infty \)[/tex].
3. Combine the effects of the leading term with the entire polynomial:
- The other terms in the polynomial, [tex]\( -26x \)[/tex] and [tex]\( -24 \)[/tex], become relatively insignificant compared to [tex]\( 2x^3 \)[/tex] for large absolute values of [tex]\( x \)[/tex].
4. Conclude the end behavior:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex].
Based on this analysis, the correct description of the end behavior of the polynomial function [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex] is:
- As [tex]\( x \rightarrow -\infty, y \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
So, the correct end behavior is:
As [tex]\( x \rightarrow -\infty, y \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
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