To determine the end behavior of the polynomial function [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex], we need to consider two key aspects: the degree of the polynomial and the leading coefficient.
1. Degree of the Polynomial:
The degree of a polynomial function is the highest power of the variable [tex]\( x \)[/tex]. For the polynomial [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex], the highest power of [tex]\( x \)[/tex] is 3. Therefore, the degree of the polynomial is [tex]\( 3 \)[/tex].
2. Leading Coefficient:
The leading coefficient is the coefficient of the term with the highest power of [tex]\( x \)[/tex]. In the polynomial [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex], the leading coefficient is [tex]\( 2 \)[/tex].
Next, we analyze the end behavior based on the degree and the leading coefficient. We use the following rules for end behavior:
- For polynomials with an odd degree:
- If the leading coefficient is positive, as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
- If the leading coefficient is negative, as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex] and as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex].
Given that the polynomial [tex]\( f(x) = 2x^3 - 26x - 24 \)[/tex] has an odd degree of 3 and a positive leading coefficient of 2, the end behavior of the graph can be described as follows:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
Thus, the correct description of the end behavior is:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
