To determine the end behavior of the polynomial function [tex]\( y = 7x^{12} - 3x^8 - 9x^4 \)[/tex], we need to focus on the highest degree term because it will dominate the behavior of the polynomial as [tex]\( x \)[/tex] approaches [tex]\(\pm\infty\)[/tex].
1. Identify the Leading Term:
The given polynomial is [tex]\( y = 7x^{12} - 3x^8 - 9x^4 \)[/tex]. The term with the highest degree is [tex]\( 7x^{12} \)[/tex].
2. Determine the Leading Term's Degree and Coefficient:
- The degree of the leading term [tex]\( 7x^{12} \)[/tex] is 12.
- The leading coefficient of [tex]\( 7x^{12} \)[/tex] is 7, which is positive.
3. Analyze the Leading Term:
- Since the degree is 12 (which is even), the behavior of the graph at both ends (as [tex]\( x \to \pm\infty \)[/tex]) will be the same.
- A positive leading coefficient (7) indicates that as [tex]\( x \to \pm\infty \)[/tex], the value of [tex]\( y \)[/tex] will also tend to positive infinity because an even-powered term with a positive coefficient grows positively in both directions.
4. End Behavior:
- As [tex]\( x \to -\infty \)[/tex], the leading term [tex]\( 7x^{12} \)[/tex] (and hence the polynomial) will go to [tex]\( +\infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], the leading term [tex]\( 7x^{12} \)[/tex] (and hence the polynomial) will also go to [tex]\( +\infty \)[/tex].
Therefore, the end behavior of the polynomial [tex]\( y = 7x^{12} - 3x^8 - 9x^4 \)[/tex] is:
- 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] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
Hence, the answer is:
[tex]\[ \boxed{4} \][/tex]
