3.3E: Graphs of Polynomial Functions (Exercises)
- Page ID
- 13891
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.
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38. 
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44.
Write a formula for each polynomial function graphed.
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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
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\(C(t)\) \(C\), 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.
15.
17.
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