3.1E: Power Functions (Exercises)

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section 3.1 EXERCISE

Find the long run behavior of each function as $x \to \infty$ and $x \to -\infty$

1. $f(x)=x^{4}$

2. $f(x)=x^{6}$

3. $f(x)=x^{3}$

4. $f(x)=x^{5}$

5. $f(x)=-x^{2}$

6. $f(x)=-x^{4}$

7. $f(x)=-x^{7}$

8. $f(x)=-x^{9}$

Find the degree and leading coefficient of each polynomial

9. $4x^{7}$

10. $5x^{6}$

11. $5-x^{2}$

12. $6+3x-4x^{3}$

13. $-2x^{4} - 3x^{2} + x-1$

14. $6x^{5} -2x^{4} + x^{2} + 3$

15. $(2x+3)(x-4)(3x+1)$

16. $(3x+1)(x+1)(4x+3)$

Find the long run behavior of each function as $x \to \infty$ and $x \to -\infty$

17. $-2x^{4} - 3x^{2} + x-1$

18. $6x^{5} -2x^{4} + x^{2} + 3$

19. $3x^{2} + x-2$

20. $-2x^{3} + x^{2} -x+3$

21. What is the maximum number of $x$-intercepts and turning points for a polynomial of degree 5?

22. What is the maximum number of $x$-intercepts and turning points for a polynomial of degree 8?

What is the least possible degree of the polynomial function shown in each graph?

23. $屏幕快照 2019-06-22 上午10.45.33.png$ 24 $屏幕快照 2019-06-22 上午10.45.56.png$ .25. $屏幕快照 2019-06-22 上午10.46.26.png$ 26. $屏幕快照 2019-06-22 上午10.46.46.png$

27. $屏幕快照 2019-06-22 上午10.47.08.png$ 28. $屏幕快照 2019-06-22 上午10.47.39.png$ 29. $屏幕快照 2019-06-22 上午10.48.00.png$ 30. $屏幕快照 2019-06-22 上午10.48.27.png$

Find the vertical and horizontal intercepts of each function.

31. $f(t)=2(t-1)(t+2)(t-3)$

32. $f(x)=3(x+1)(x-4)(x+5)$

33. $g(n)=-2(3n-1)(2n+1)$

34. $k(u)=-3(4-n)(4n+3)$

Answer

1. As $x \to \infty$, $f(x) \to \infty$ As $x \to -\infty$, $f(x) \to \infty$

3. As $x \to \infty$, $f(x) \to \infty$ As $x \to -\infty$, $f(x) \to -\infty$

5. As $x \to \infty$, $f(x) \to -\infty$ As $x \to -\infty$, $f(x) \to -\infty$

7. As $x \to \infty$, $f(x) \to -\infty$ As $x \to -\infty$, $f(x) \to \infty$

9. $7^{\text{th}}$ Degree, Leading coefficient 4

11. $2^{\text{nd}}$ Degree, Leading coefficient -1

13. $4^{\text{th}}$ Degree, Leading coefficient -2

15. $3^{\text{rd}}$ Degree, Leading coefficient 6

17. As $x \to \infty$, $f(x) \to -\infty$ As $x \to -\infty$, $f(x) \to -\infty$

19. As $x \to \infty$, $f(x) \to \infty$ As $x \to -\infty$, $f(x) \to \infty$

21. intercepts: 5, turning points: 4

23. 3

25. 5

27. 3

29. 5

31. Horizontal Intercepts (1, 0), (-2, 0), (3, 0) Vertical Intercept (0, 12)

33. Horizontal Intercepts (1/3, 0) (-1/2. 0) Vertical Intercept (0, 2)