To determine the end behavior of the polynomial function [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex], we focus on the term with the highest degree, which is the leading term. In this case, the leading term is [tex]\( 3x^6 \)[/tex].
Here's a step-by-step explanation of how to understand the end behavior of this polynomial:
1. Degree of the Polynomial:
The highest degree of the polynomial [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex] is 6, which is even.
2. Leading Coefficient:
The leading coefficient (the coefficient of the highest degree term) is 3, which is positive.
3. Behavior Based on Degree and Leading Coefficient:
- For polynomials with an even degree, the end behavior is the same in both directions.
- If the leading coefficient is positive, then as [tex]\( x \)[/tex] goes to [tex]\( +\infty \)[/tex], [tex]\( y \)[/tex] goes to [tex]\( +\infty \)[/tex], and as [tex]\( x \)[/tex] goes to [tex]\( -\infty \)[/tex], [tex]\( y \)[/tex] also goes to [tex]\( +\infty \)[/tex].
Given these points, we can conclude the end behavior of the polynomial:
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 of the polynomial [tex]\( f(x) = 3x^6 + 30x^5 + 75x^4 \)[/tex] is:
As [tex]\( x \rightarrow-\infty, y \rightarrow \infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
The statement that fits this behavior is:
As [tex]\( x \rightarrow-\infty, y \rightarrow \infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
Therefore, the answer is:
1. As [tex]\( x \rightarrow-\infty, y \rightarrow \infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
