Get expert advice and community support for your questions on IDNLearn.com. Join our interactive Q&A platform to receive prompt and accurate responses from experienced professionals in various fields.
Sagot :
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]
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]
Your participation means a lot to us. Keep sharing information and solutions. This community grows thanks to the amazing contributions from members like you. IDNLearn.com is your source for precise answers. Thank you for visiting, and we look forward to helping you again soon.