To determine the end behavior of the polynomial function [tex]\( y = 7x^{12} - 3x^8 - 9x^4 \)[/tex], we need to focus on the term with the highest degree, as it will dominate the behavior of the polynomial for very large and very small values of [tex]\( x \)[/tex].
1. Identify the term with the highest degree:
The term with the highest degree in the polynomial is [tex]\( 7x^{12} \)[/tex].
2. Analyze the coefficient and exponent of the highest degree term:
- The coefficient of [tex]\( x^{12} \)[/tex] is [tex]\( 7 \)[/tex], which is positive.
- The exponent of [tex]\( x^{12} \)[/tex] is [tex]\( 12 \)[/tex], which is even.
When analyzing polynomial end behavior:
- If the exponent is even and the leading coefficient is positive, the polynomial will tend to [tex]\( \infty \)[/tex] (positive infinity) as [tex]\( x \)[/tex] approaches both [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex].
- If the exponent is even and the leading coefficient is negative, the polynomial will tend to [tex]\( -\infty \)[/tex] (negative infinity) as [tex]\( x \)[/tex] approaches both [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex].
- If the exponent is odd and the leading coefficient is positive, the polynomial will tend to [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex].
- If the exponent is odd and the leading coefficient is negative, the polynomial will tend to [tex]\( -\infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] and [tex]\( \infty \)[/tex] as [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex].
Given that [tex]\( 12 \)[/tex] is even and [tex]\( 7 \)[/tex] is positive, we conclude:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
Thus, the end behavior of the graph of the polynomial function [tex]\( y = 7x^{12} - 3x^8 - 9x^4 \)[/tex] is
[tex]\[ \text{As } x \rightarrow -\infty, y \rightarrow \infty \text{ and as } x \rightarrow \infty, y \rightarrow \infty. \][/tex]
