Answer :
To determine the end behavior of the polynomial function \( f(x) = 2x^3 - 26x - 24 \), we need to focus on the term with the highest degree, which dominates the behavior of the function as \( x \) approaches very large positive and negative values.
For the polynomial \( f(x) = 2x^3 - 26x - 24 \):
1. Identify the leading term:
The leading term of the polynomial is \( 2x^3 \).
2. Analyze what happens as \( x \) approaches \( -\infty \) (negative infinity):
- For large negative values of \( x \), the cubic term \( 2x^3 \) will dominate.
- When \( x \) is a large negative number, \( x^3 \) will be negative (since an odd degree of a negative number is negative).
- Therefore, \( 2x^3 \) will be a large negative number.
- Thus, as \( x \rightarrow -\infty \), \( 2x^3 \rightarrow -\infty \).
3. Analyze what happens as \( x \) approaches \( \infty \) (positive infinity):
- For large positive values of \( x \), the cubic term \( 2x^3 \) will once again dominate.
- When \( x \) is a large positive number, \( x^3 \) will be positive.
- Therefore, \( 2x^3 \) will be a large positive number.
- Thus, as \( x \rightarrow \infty \), \( 2x^3 \rightarrow \infty \).
Given this analysis, the end behavior of the polynomial function \( f(x) \) is:
- As \( x \rightarrow -\infty \), \( y \rightarrow -\infty \).
- As \( x \rightarrow \infty \), \( y \rightarrow \infty \).
Hence, the correct answer is:
As [tex]\( x \rightarrow -\infty, y \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
For the polynomial \( f(x) = 2x^3 - 26x - 24 \):
1. Identify the leading term:
The leading term of the polynomial is \( 2x^3 \).
2. Analyze what happens as \( x \) approaches \( -\infty \) (negative infinity):
- For large negative values of \( x \), the cubic term \( 2x^3 \) will dominate.
- When \( x \) is a large negative number, \( x^3 \) will be negative (since an odd degree of a negative number is negative).
- Therefore, \( 2x^3 \) will be a large negative number.
- Thus, as \( x \rightarrow -\infty \), \( 2x^3 \rightarrow -\infty \).
3. Analyze what happens as \( x \) approaches \( \infty \) (positive infinity):
- For large positive values of \( x \), the cubic term \( 2x^3 \) will once again dominate.
- When \( x \) is a large positive number, \( x^3 \) will be positive.
- Therefore, \( 2x^3 \) will be a large positive number.
- Thus, as \( x \rightarrow \infty \), \( 2x^3 \rightarrow \infty \).
Given this analysis, the end behavior of the polynomial function \( f(x) \) is:
- As \( x \rightarrow -\infty \), \( y \rightarrow -\infty \).
- As \( x \rightarrow \infty \), \( y \rightarrow \infty \).
Hence, the correct answer is:
As [tex]\( x \rightarrow -\infty, y \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
