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Mathematics LibreTexts

3.3E: Graphs of Polynomial Functions (Exercises)

  • Page ID
    13891
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    section 3.3 exercise

    Find the \(C\) and \(t\) intercepts of each function.

    1. \(C\left(t\right)=2\left(t-4\right)\left(t+1\right)(t-6)\)

    2. \(C\left(t\right)=3\left(t+2\right)\left(t-3\right)(t+5)\)

    3. \(C\left(t\right)=4t\left(t-2\right)^{2} (t+1)\)

    4. \(C\left(t\right)=2t\left(t-3\right)\left(t+1\right)^{2}\)

    5. \(C\left(t\right)=2t^{4} -8t^{3} +6t^{2}\)

    6. \(C\left(t\right)=4t^{4} +12t^{3} -40t^{2}\)

    Use your calculator or other graphing technology to solve graphically for the zeros of the function.

    7. \(f\left(x\right)=x^{3} -7x^{2} +4x+30\)

    8. \(g\left(x\right)=x^{3} -6x^{2} +x+28\)

    Find the long run behavior of each function as \(t \to \infty\) and \(t \to -\infty\)

    9. \(h\left(t\right)=3\left(t-5\right)^{3} \left(t-3\right)^{3} (t-2)\)

    10. \(k\left(t\right)=2\left(t-3\right)^{2} \left(t+1\right)^{3} (t+2)\)

    11. \(p\left(t\right)=-2t\left(t-1\right)\left(3-t\right)^{2}\)

    12. \(q\left(t\right)=-4t\left(2-t\right)\left(t+1\right)^{3}\)

    Sketch a graph of each equation.

    13. \(f\left(x\right)=\left(x+3\right)^{2} (x-2)\)

    14. \(g\left(x\right)=\left(x+4\right)\left(x-1\right)^{2}\)

    15. \(h\left(x\right)=\left(x-1\right)^{3} \left(x+3\right)^{2}\)

    16. \(k\left(x\right)=\left(x-3\right)^{3} \left(x-2\right)^{2}\)

    17. \(m\left(x\right)=-2x\left(x-1\right)(x+3)\)

    18. \(n\left(x\right)=-3x\left(x+2\right)(x-4)\)

    Solve each inequality.

    19. \(\left(x-3\right)\left(x-2\right)^{2} >0\)

    20. \(\left(x-5\right)\left(x+1\right)^{2} >0\)

    21. \(\left(x-1\right)\left(x+2\right)\left(x-3\right)<0\)

    22. \(\left(x-4\right)\left(x+3\right)\left(x+6\right)<0\)

    Find the domain of each function.

    23. \(f\left(x\right)=\sqrt{-42+19x-2x^{2} }\)

    24. \(g\left(x\right)=\sqrt{28-17x-3x^{2} }\)

    25. \(h\left(x\right)=\sqrt{4-5x+x^{2} }\)

    26. \(k\left(x\right)=\sqrt{2+7x+3x^{2} }\)

    27. \(n\left(x\right)=\sqrt{\left(x-3\right)\left(x+2\right)^{2} }\)

    28. \(m\left(x\right)=\sqrt{\left(x-1\right)^{2} (x+3)}\)

    29. \(p\left(t\right)=\dfrac{1}{t^{2} +2t-8}\)

    30. \(q\left(t\right)=\dfrac{4}{x^{2} -4x-5}\)

    Write an equation for a polynomial the given features.

    31. Degree 3. Zeros at \(x\) = -2, \(x\) = 1, and \(x\) = 3. Vertical intercept at (0, -4)

    32. Degree 3. Zeros at \(x\) = -5, \(x\) = -2, and \(x\) = 1. Vertical intercept at (0, 6)

    33. Degree 5. Roots of multiplicity 2 at \(x\) = 3 and \(x\) = 1, and a root of multiplicity 1 at \(x\) = -3. Vertical intercept at (0, 9)

    34. Degree 4. Root of multiplicity 2 at \(x\) = 4, and a roots of multiplicity 1 at \(x\) = 1 and \(x\) = -2. Vertical intercept at (0, -3)

    35. Degree 5. Double zero at \(x\) = 1, and triple zero at \(x\) = 3. Passes through the point (2, 15)

    36. Degree 5. Single zero at \(x\) = -2 and \(x\) = 3, and triple zero at \(x\) = 1. Passes through the point (2, 4)

    Write a formula for each polynomial function graphed.

    37. 屏幕快照 2019-06-23 上午3.24.15.png38. 屏幕快照 2019-06-23 上午3.24.35.png39.屏幕快照 2019-06-23 上午3.24.57.png

    40. 屏幕快照 2019-06-23 上午3.25.18.png41. 屏幕快照 2019-06-23 上午3.25.44.png42. 屏幕快照 2019-06-23 上午3.26.12.png

    43. 屏幕快照 2019-06-23 上午3.26.29.png44.屏幕快照 2019-06-23 上午3.26.46.png

    Write a formula for each polynomial function graphed.

    45. 屏幕快照 2019-06-23 上午5.32.03.png46.屏幕快照 2019-06-23 上午5.32.31.png

    47. 屏幕快照 2019-06-23 上午5.32.47.png48. 屏幕快照 2019-06-23 上午5.33.17.png

    49.屏幕快照 2019-06-23 上午5.33.39.png 50. 屏幕快照 2019-06-23 上午5.34.00.png

    51. A rectangle is inscribed with its base on the \(x\) axis and its upper corners on the parabola \(y=5-x^{2}\). What are the dimensions of such a rectangle that has the greatest possible area?

    52. A rectangle is inscribed with its base on the \(x\) axis and its upper corners on the curve \(y=16-x^{4}\). What are the dimensions of such a rectangle that has the greatest possible area?

    Answer
    \(C(t)\) \(C\), intercepts \(t\), intercepts
    1. (0, 48) (4, 0), (-1, 0), (6, 0)
    3. (0, 0) (0, 0), (2, 0), (-1, 0)
    5. (0, 0) (0, 0), (1, 0), (3, 0)

    7. (-1.646, 0) (3.646, 0) (5, 0)

    9. As \(t \to \infty\), \(h(t) \to \infty\) \(t \to -\infty\), \(h(t) \to -\infty\)

    11. As \(t \to \infty\), \(p(t) \to -\infty\) \(t \to -\infty\), \(p(t) \to -\infty\)

    13. Screen Shot 2019-10-03 at 6.05.35 PM.png

    15. Screen Shot 2019-10-03 at 6.05.54 PM.png

    17. Screen Shot 2019-10-03 at 6.06.33 PM.png

    19. \((3, \infty)\)

    21. \((-\infty, -2) \cup (1, 3)\)

    23. [3, 5, 6]

    25. \((-\infty, 1] \cup [4, \infty)\)

    27. \([-2, -2] \cup [3, \infty)\)

    29. \((-\infty, -4) \cup (-4, 2) \cup (2, \infty)\)

    31. \(y = -\dfrac{2}{3} (x + 2) (x - 1) (x - 3)\)

    33. \(y = \dfrac{1}{3} (x - 1)^2 (x - 3)^2 (x + 3)\)

    35. \(y = -15(x - 1)^2 (x - 3)^2\)

    37. \(y = \dfrac{1}{2} (x + 2)(x - 1) (x - 3)\)

    39. \(y = -(x + 1)^2 (x - 2)\)

    41. \(y = -\dfrac{1}{24} (x + 3)(x + 2) (x - 2) (x - 4)\)

    43. \(y = \dfrac{1}{24} (x + 4) (x + 2) (x - 3)^2\)

    45. \(y = \dfrac{1}{12} (x + 2)^2 (x - 3)^2\)

    47. \(y = \dfrac{1}{6} (x + 3) (x + 2) (x - 1)^3\)

    49. \(y = -\dfrac{1}{16} (x + 3)(x + 1) (x - 2)^2 (x - 4)\)

    51. Base 2.58, Height 3.336