Answer :
To determine the end behavior of the polynomial function [tex]\( g(x) = -10x^7 - 9x^5 + 7x^3 - 8x \)[/tex], we need to focus on the term with the highest degree because it dominates the behavior of the polynomial as [tex]\( x \)[/tex] goes to positive or negative infinity.
1. Identify the Leading Term:
The term with the highest degree in the polynomial is [tex]\(-10x^7\)[/tex]. This term will significantly affect the end behavior of the polynomial.
2. Analyze the Leading Term:
- The degree of the leading term is 7, which is an odd number.
- The coefficient of the leading term is [tex]\(-10\)[/tex], which is negative.
3. End Behavior Analysis:
For a polynomial where the leading term is of the form [tex]\(-ax^n\)[/tex] with [tex]\(n\)[/tex] being odd and [tex]\(a\)[/tex] being positive:
- As [tex]\(x\)[/tex] approaches positive infinity ([tex]\(x \to \infty\)[/tex]), [tex]\(-10x^7\)[/tex] becomes increasingly negative. Therefore, [tex]\(g(x) \to -\infty\)[/tex].
- As [tex]\(x\)[/tex] approaches negative infinity ([tex]\(x \to -\infty\)[/tex]), [tex]\(x^7\)[/tex] would still be negative since it's an odd power, making [tex]\(-10x^7\)[/tex] positive. Therefore, [tex]\(g(x) \to \infty\)[/tex].
4. Conclusion:
Based on the analysis, the end behavior of the graph of [tex]\( g(x) \)[/tex] is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
This means the graph of [tex]\( g(x) \)[/tex] will fall to negative infinity as [tex]\( x \)[/tex] moves to the right and rise to positive infinity as [tex]\( x \)[/tex] moves to the left.
1. Identify the Leading Term:
The term with the highest degree in the polynomial is [tex]\(-10x^7\)[/tex]. This term will significantly affect the end behavior of the polynomial.
2. Analyze the Leading Term:
- The degree of the leading term is 7, which is an odd number.
- The coefficient of the leading term is [tex]\(-10\)[/tex], which is negative.
3. End Behavior Analysis:
For a polynomial where the leading term is of the form [tex]\(-ax^n\)[/tex] with [tex]\(n\)[/tex] being odd and [tex]\(a\)[/tex] being positive:
- As [tex]\(x\)[/tex] approaches positive infinity ([tex]\(x \to \infty\)[/tex]), [tex]\(-10x^7\)[/tex] becomes increasingly negative. Therefore, [tex]\(g(x) \to -\infty\)[/tex].
- As [tex]\(x\)[/tex] approaches negative infinity ([tex]\(x \to -\infty\)[/tex]), [tex]\(x^7\)[/tex] would still be negative since it's an odd power, making [tex]\(-10x^7\)[/tex] positive. Therefore, [tex]\(g(x) \to \infty\)[/tex].
4. Conclusion:
Based on the analysis, the end behavior of the graph of [tex]\( g(x) \)[/tex] is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( g(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
This means the graph of [tex]\( g(x) \)[/tex] will fall to negative infinity as [tex]\( x \)[/tex] moves to the right and rise to positive infinity as [tex]\( x \)[/tex] moves to the left.
