To determine the end behavior of the polynomial function [tex]\( y = 7x^{12} - 3x^3 - 9x^4 \)[/tex], we need to consider the term with the highest degree, which will dominate the behavior of the function as [tex]\( x \)[/tex] approaches infinity or negative infinity.
The highest degree term in the polynomial is [tex]\( 7x^{12} \)[/tex]. Let's analyze this term:
1. As [tex]\( x \rightarrow \infty \)[/tex]:
- When [tex]\( x \)[/tex] becomes very large (approaching infinity), the term [tex]\( 7x^{12} \)[/tex] will dominate because it grows much faster than the other terms [tex]\( -3x^3 \)[/tex] and [tex]\( -9x^4 \)[/tex].
- Since [tex]\( x^{12} \)[/tex] is always positive for positive [tex]\( x \)[/tex] and multiplying by the positive coefficient 7, the polynomial [tex]\( 7x^{12} \)[/tex] grows to [tex]\( \infty \)[/tex].
- Therefore, [tex]\( y \rightarrow \infty \)[/tex].
2. As [tex]\( x \rightarrow -\infty \)[/tex]:
- When [tex]\( x \)[/tex] becomes very large in the negative direction (approaching negative infinity), the term [tex]\( 7x^{12} \)[/tex] will still dominate.
- Since [tex]\( x^{12} \)[/tex] remains positive for negative [tex]\( x \)[/tex] (because an even exponent of a negative number is positive), multiplying by the positive coefficient 7, the polynomial [tex]\( 7x^{12} \)[/tex] still grows to [tex]\( \infty \)[/tex].
- Therefore, [tex]\( y \rightarrow \infty \)[/tex].
Given this analysis, the end behavior of the graph of the polynomial function [tex]\( y = 7x^{12} - 3x^3 - 9x^4 \)[/tex] is:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
Thus, the correct statement regarding the end behavior is:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex] and as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
