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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.
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