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Mathematics LibreTexts

3.1.1E: Power Functions (Exercises)

  • Page ID
    13889
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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.png24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.png26.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.png28.屏幕快照 2019-06-22 上午10.47.39.png29.屏幕快照 2019-06-22 上午10.48.00.png30. 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)