Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
Sagot :
Sure! Let's analyze how adding the terms [tex]\(-x^8\)[/tex] and [tex]\(5x^7\)[/tex] to the polynomial [tex]\(y = -2x^7 + 5x^6 - 24\)[/tex] changes the end behavior of the graph.
### Adding the Term [tex]\(-x^8\)[/tex]
1. Identify the Highest Degree Term:
- In the polynomial [tex]\(y = -2x^7 + 5x^6 - 24\)[/tex], the term with the highest degree is [tex]\(-2x^7\)[/tex].
- When we add [tex]\(-x^8\)[/tex], the highest degree term becomes [tex]\(-x^8\)[/tex].
2. Determine the End Behavior:
- The end behavior of a polynomial is dictated by its highest degree term. For [tex]\(-x^8\)[/tex], the highest degree term is [tex]\(-x^8\)[/tex].
- As [tex]\(x\)[/tex] approaches positive infinity ([tex]\(x \to +\infty\)[/tex]):
- The term [tex]\(-x^8\)[/tex] dominates, and because the coefficient of [tex]\(x^8\)[/tex] is negative, the value of [tex]\(y\)[/tex] will approach negative infinity ([tex]\(y \to -\infty\)[/tex]).
- As [tex]\(x\)[/tex] approaches negative infinity ([tex]\(x \to -\infty\)[/tex]):
- Similarly, the term [tex]\(-x^8\)[/tex] dominates, and because the coefficient remains negative, the value of [tex]\(y\)[/tex] will also approach negative infinity ([tex]\(y \to -\infty\)[/tex]).
So, the end behavior of the graph when adding [tex]\(-x^8\)[/tex] will be negative infinity as [tex]\(x\)[/tex] approaches both positive and negative infinity.
### Adding the Term [tex]\(5x^7\)[/tex]
1. Identify the Highest Degree Term:
- In the original polynomial [tex]\(y = -2x^7 + 5x^6 - 24\)[/tex], the term with the highest degree is [tex]\(-2x^7\)[/tex].
- When we add [tex]\(5x^7\)[/tex], the highest degree term changes as follows:
[tex]\[ -2x^7 + 5x^7 = 3x^7 \][/tex]
- So after adding [tex]\(5x^7\)[/tex], the highest degree term of the new polynomial will be [tex]\(3x^7\)[/tex].
2. Determine the End Behavior:
- For [tex]\(3x^7\)[/tex] as the highest degree term:
- As [tex]\(x\)[/tex] approaches positive infinity ([tex]\(x \to +\infty\)[/tex]):
- The term [tex]\(3x^7\)[/tex] dominates, and because the coefficient of [tex]\(x^7\)[/tex] is positive, the value of [tex]\(y\)[/tex] will approach positive infinity ([tex]\(y \to +\infty\)[/tex]).
- As [tex]\(x\)[/tex] approaches negative infinity ([tex]\(x \to -\infty\)[/tex]):
- The term [tex]\(3x^7\)[/tex] dominates, and because the coefficient is positive but the exponent is odd, the value of [tex]\(y\)[/tex] will approach negative infinity ([tex]\(y \to -\infty\)[/tex]).
So, the end behavior of the graph when adding [tex]\(5x^7\)[/tex] will be positive infinity as [tex]\(x\)[/tex] approaches positive infinity, and negative infinity as [tex]\(x\)[/tex] approaches negative infinity.
### Summary:
- Adding [tex]\(-x^8\)[/tex]: The end behavior will be negative infinity as [tex]\(x\)[/tex] approaches either positive or negative infinity.
- Adding [tex]\(5x^7\)[/tex]: The end behavior will be positive infinity as [tex]\(x\)[/tex] approaches positive infinity, and negative infinity as [tex]\(x\)[/tex] approaches negative infinity.
### Adding the Term [tex]\(-x^8\)[/tex]
1. Identify the Highest Degree Term:
- In the polynomial [tex]\(y = -2x^7 + 5x^6 - 24\)[/tex], the term with the highest degree is [tex]\(-2x^7\)[/tex].
- When we add [tex]\(-x^8\)[/tex], the highest degree term becomes [tex]\(-x^8\)[/tex].
2. Determine the End Behavior:
- The end behavior of a polynomial is dictated by its highest degree term. For [tex]\(-x^8\)[/tex], the highest degree term is [tex]\(-x^8\)[/tex].
- As [tex]\(x\)[/tex] approaches positive infinity ([tex]\(x \to +\infty\)[/tex]):
- The term [tex]\(-x^8\)[/tex] dominates, and because the coefficient of [tex]\(x^8\)[/tex] is negative, the value of [tex]\(y\)[/tex] will approach negative infinity ([tex]\(y \to -\infty\)[/tex]).
- As [tex]\(x\)[/tex] approaches negative infinity ([tex]\(x \to -\infty\)[/tex]):
- Similarly, the term [tex]\(-x^8\)[/tex] dominates, and because the coefficient remains negative, the value of [tex]\(y\)[/tex] will also approach negative infinity ([tex]\(y \to -\infty\)[/tex]).
So, the end behavior of the graph when adding [tex]\(-x^8\)[/tex] will be negative infinity as [tex]\(x\)[/tex] approaches both positive and negative infinity.
### Adding the Term [tex]\(5x^7\)[/tex]
1. Identify the Highest Degree Term:
- In the original polynomial [tex]\(y = -2x^7 + 5x^6 - 24\)[/tex], the term with the highest degree is [tex]\(-2x^7\)[/tex].
- When we add [tex]\(5x^7\)[/tex], the highest degree term changes as follows:
[tex]\[ -2x^7 + 5x^7 = 3x^7 \][/tex]
- So after adding [tex]\(5x^7\)[/tex], the highest degree term of the new polynomial will be [tex]\(3x^7\)[/tex].
2. Determine the End Behavior:
- For [tex]\(3x^7\)[/tex] as the highest degree term:
- As [tex]\(x\)[/tex] approaches positive infinity ([tex]\(x \to +\infty\)[/tex]):
- The term [tex]\(3x^7\)[/tex] dominates, and because the coefficient of [tex]\(x^7\)[/tex] is positive, the value of [tex]\(y\)[/tex] will approach positive infinity ([tex]\(y \to +\infty\)[/tex]).
- As [tex]\(x\)[/tex] approaches negative infinity ([tex]\(x \to -\infty\)[/tex]):
- The term [tex]\(3x^7\)[/tex] dominates, and because the coefficient is positive but the exponent is odd, the value of [tex]\(y\)[/tex] will approach negative infinity ([tex]\(y \to -\infty\)[/tex]).
So, the end behavior of the graph when adding [tex]\(5x^7\)[/tex] will be positive infinity as [tex]\(x\)[/tex] approaches positive infinity, and negative infinity as [tex]\(x\)[/tex] approaches negative infinity.
### Summary:
- Adding [tex]\(-x^8\)[/tex]: The end behavior will be negative infinity as [tex]\(x\)[/tex] approaches either positive or negative infinity.
- Adding [tex]\(5x^7\)[/tex]: The end behavior will be positive infinity as [tex]\(x\)[/tex] approaches positive infinity, and negative infinity as [tex]\(x\)[/tex] approaches negative infinity.
Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Thank you for choosing Westonci.ca as your information source. We look forward to your next visit.
