describe the end behavior of each function.
1) $f(x) = -2x^2 - 8x - 7$
2) $f(x) = -x^4 - 2x^3 + x^2$
3) $f(x) = x^3 + 5x^2 + 3x - 3$
4) $f(x) = x^4 - 2x^2 - x - 2$
5) $f(x) = x^2 - 2x - 2$
6) $f(x) = -2x^2 + 8x - 4$
7) $f(x) = x^3 - 12x^2 + 45x - 55$
8) $f(x) = -x^3 + 3x^2 - 5$
9) $f(x) = x^4 - 2x^2 + x - 3$
10) $f(x) = x^2$
11) $f(x) = x^5 - 4x^3 + x - 1$
12) $f(x) = x^2 - 4x - 1$
13) $f(x) = -x^3 + x^2 - 4$
14) $f(x) = -2x^2 + 16x - 30$
15) $f(x) = -x^5 + 4x^3 - 3x + 3$
16) $f(x) = -x^2 - 4x - 1$

Question

Accepted Answer

# Explanation: ## Step1: Recall End Behavior Rules For a polynomial $ f(x) = a_nx^n + a_{n - 1}x^{n - 1}+\dots+a_1x + a_0 $, the end behavior is determined by the leading term $ a_nx^n $: - If $ n $ is even: - If $ a_n>0 $, as $ x\to\pm\infty $, $ f(x)\to+\infty $. - If $ a_n<0 $, as $ x\to\pm\infty $, $ f(x)\to-\infty $. - If $ n $ is odd: - If $ a_n>0 $, as $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $. - If $ a_n<0 $, as $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $. ## Step2: Analyze $ f(x)=-2x^2 - 8x - 7 $ (Problem 1) - Leading term: $ -2x^2 $, $ n = 2 $ (even), $ a_n=-2<0 $. - So, as $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $. ## Step3: Analyze $ f(x)=-x^4 - 2x^3 + x^2 $ (Problem 2) - Leading term: $ -x^4 $, $ n = 4 $ (even), $ a_n=-1<0 $. - So, as $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $. ## Step4: Analyze $ f(x)=x^3 + 5x^2 + 3x - 3 $ (Problem 3) - Leading term: $ x^3 $, $ n = 3 $ (odd), $ a_n=1>0 $. - So, as $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $. ## Step5: Analyze $ f(x)=x^4 - 2x^2 - x - 2 $ (Problem 4) - Leading term: $ x^4 $, $ n = 4 $ (even), $ a_n=1>0 $. - So, as $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $. ## Step6: Analyze $ f(x)=x^2 - 2x - 2 $ (Problem 5) - Leading term: $ x^2 $, $ n = 2 $ (even), $ a_n=1>0 $. - So, as $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $. ## Step7: Analyze $ f(x)=-2x^2 + 8x - 4 $ (Problem 6) - Leading term: $ -2x^2 $, $ n = 2 $ (even), $ a_n=-2<0 $. - So, as $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $. ## Step8: Analyze $ f(x)=x^3 - 12x^2 + 45x - 55 $ (Problem 7) - Leading term: $ x^3 $, $ n = 3 $ (odd), $ a_n=1>0 $. - So, as $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $. ## Step9: Analyze $ f(x)=-x^3 + 3x^2 - 5 $ (Problem 8) - Leading term: $ -x^3 $, $ n = 3 $ (odd), $ a_n=-1<0 $. - So, as $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $. ## Step10: Analyze $ f(x)=x^4 - 2x^2 + x - 3 $ (Problem 9) - Leading term: $ x^4 $, $ n = 4 $ (even), $ a_n=1>0 $. - So, as $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $. ## Step11: Analyze $ f(x)=x^2 $ (Problem 10) - Leading term: $ x^2 $, $ n = 2 $ (even), $ a_n=1>0 $. - So, as $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $. ## Step12: Analyze $ f(x)=x^5 - 4x^3 + x - 1 $ (Problem 11) - Leading term: $ x^5 $, $ n = 5 $ (odd), $ a_n=1>0 $. - So, as $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $. ## Step13: Analyze $ f(x)=x^2 - 4x - 1 $ (Problem 12) - Leading term: $ x^2 $, $ n = 2 $ (even), $ a_n=1>0 $. - So, as $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $. ## Step14: Analyze $ f(x)=-x^3 + x^2 - 4 $ (Problem 13) - Leading term: $ -x^3 $, $ n = 3 $ (odd), $ a_n=-1<0 $. - So, as $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $. ## Step15: Analyze $ f(x)=-2x^2 + 16x - 30 $ (Problem 14) - Leading term: $ -2x^2 $, $ n = 2 $ (even), $ a_n=-2<0 $. - So, as $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $. ## Step16: Analyze $ f(x)=-x^5 + 4x^3 - 3x + 3 $ (Problem 15) - Leading term: $ -x^5 $, $ n = 5 $ (odd), $ a_n=-1<0 $. - So, as $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $. ## Step17: Analyze $ f(x)=-x^2 - 4x - 1 $ (Problem 16) - Leading term: $ -x^2 $, $ n = 2 $ (even), $ a_n=-1<0 $. - So, as $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $. # Answer: 1. As $ x\to\pm\infty $, $ f(x)\to-\infty $ 2. As $ x\to\pm\infty $, $ f(x)\to-\infty $ 3. As $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $ 4. As $ x\to\pm\infty $, $ f(x)\to+\infty $ 5. As $ x\to\pm\infty $, $ f(x)\to+\infty $ 6. As $ x\to\pm\infty $, $ f(x)\to-\infty $ 7. As $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $ 8. As $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $ 9. As $ x\to\pm\infty $, $ f(x)\to+\infty $ 10. As $ x\to\pm\infty $, $ f(x)\to+\infty $ 11. As $ x\to+\infty $, $ f(x)\to+\infty $; as $ x\to-\infty $, $ f(x)\to-\infty $ 12. As $ x\to\pm\infty $, $ f(x)\to+\infty $ 13. As $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $ 14. As $ x\to\pm\infty $, $ f(x)\to-\infty $ 15. As $ x\to+\infty $, $ f(x)\to-\infty $; as $ x\to-\infty $, $ f(x)\to+\infty $ 16. As $ x\to\pm\infty $, $ f(x)\to-\infty $

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QUESTION IMAGE

Question

Explanation:

Step1: Recall End Behavior Rules

Step2: Analyze \( f(x)=-2x^2 - 8x - 7 \) (Problem 1)

Step3: Analyze \( f(x)=-x^4 - 2x^3 + x^2 \) (Problem 2)

Step4: Analyze \( f(x)=x^3 + 5x^2 + 3x - 3 \) (Problem 3)

Step5: Analyze \( f(x)=x^4 - 2x^2 - x - 2 \) (Problem 4)

Step6: Analyze \( f(x)=x^2 - 2x - 2 \) (Problem 5)

Step7: Analyze \( f(x)=-2x^2 + 8x - 4 \) (Problem 6)

Step8: Analyze \( f(x)=x^3 - 12x^2 + 45x - 55 \) (Problem 7)

Step9: Analyze \( f(x)=-x^3 + 3x^2 - 5 \) (Problem 8)

Step10: Analyze \( f(x)=x^4 - 2x^2 + x - 3 \) (Problem 9)

Step11: Analyze \( f(x)=x^2 \) (Problem 10)

Step12: Analyze \( f(x)=x^5 - 4x^3 + x - 1 \) (Problem 11)

Step13: Analyze \( f(x)=x^2 - 4x - 1 \) (Problem 12)

Step14: Analyze \( f(x)=-x^3 + x^2 - 4 \) (Problem 13)

Step15: Analyze \( f(x)=-2x^2 + 16x - 30 \) (Problem 14)

Step16:…

Answer: