Section Exercises
1. Explain the difference between the coefficient of a power function and its degree. 2. If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function? 3. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive. 4. What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph? 5. What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As [latex]x\to -\infty ,f\left(x\right)\to -\infty \\[/latex] and as [latex]x\to \infty ,f\left(x\right)\to -\infty \\[/latex]. For the following exercises, identify the function as a power function, a polynomial function, or neither. 6. [latex]f\left(x\right)={x}^{5}\\[/latex] 7. [latex]f\left(x\right)={\left({x}^{2}\right)}^{3}\\[/latex] 8. [latex]f\left(x\right)=x-{x}^{4}\\[/latex] 9. [latex]f\left(x\right)=\frac{{x}^{2}}{{x}^{2}-1}\\[/latex] 10. [latex]f\left(x\right)=2x\left(x+2\right){\left(x - 1\right)}^{2}\\[/latex] 11. [latex]f\left(x\right)={3}^{x+1}\\[/latex] For the following exercises, find the degree and leading coefficient for the given polynomial. 12. [latex]-3x{}^{4}\\[/latex] 13. [latex]7 - 2{x}^{2}\\[/latex] 14. [latex]-2{x}^{2}- 3{x}^{5}+ x - 6 \\[/latex] 15. [latex]x\left(4-{x}^{2}\right)\left(2x+1\right)\\[/latex] 16. [latex]{x}^{2}{\left(2x - 3\right)}^{2}\\[/latex] For the following exercises, determine the end behavior of the functions. 17. [latex]f\left(x\right)={x}^{4}\\[/latex] 18. [latex]f\left(x\right)={x}^{3}\\[/latex] 19. [latex]f\left(x\right)=-{x}^{4}\\[/latex] 20. [latex]f\left(x\right)=-{x}^{9}\\[/latex] 21. [latex]f\left(x\right)=-2{x}^{4}- 3{x}^{2}+ x - 1\\[/latex] 22. [latex]f\left(x\right)=3{x}^{2}+ x - 2\\[/latex] 23. [latex]f\left(x\right)={x}^{2}\left(2{x}^{3}-x+1\right)\\[/latex] 24. [latex]f\left(x\right)={\left(2-x\right)}^{7}\\[/latex] For the following exercises, find the intercepts of the functions. 25. [latex]f\left(t\right)=2\left(t - 1\right)\left(t+2\right)\left(t - 3\right)\\[/latex] 26. [latex]g\left(n\right)=-2\left(3n - 1\right)\left(2n+1\right)\\[/latex] 27. [latex]f\left(x\right)={x}^{4}-16\\[/latex] 28. [latex]f\left(x\right)={x}^{3}+27\\[/latex] 29. [latex]f\left(x\right)=x\left({x}^{2}-2x - 8\right)\\[/latex] 30. [latex]f\left(x\right)=\left(x+3\right)\left(4{x}^{2}-1\right)\\[/latex] For the following exercises, determine the least possible degree of the polynomial function shown. 31.
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For the following exercises, determine whether the graph of the function provided is a graph of a polynomial function. If so, determine the number of turning points and the least possible degree for the function.
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For the following exercises, make a table to confirm the end behavior of the function.
46. [latex]f\left(x\right)=-{x}^{3}\\[/latex]
47. [latex]f\left(x\right)={x}^{4}-5{x}^{2}\\[/latex]
48. [latex]f\left(x\right)={x}^{2}{\left(1-x\right)}^{2}\\[/latex]
49. [latex]f\left(x\right)=\left(x - 1\right)\left(x - 2\right)\left(3-x\right)\\[/latex]
50. [latex]f\left(x\right)=\frac{{x}^{5}}{10}-{x}^{4}\\[/latex]
For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.
51. [latex]f\left(x\right)={x}^{3}\left(x - 2\right)\\[/latex]
52. [latex]f\left(x\right)=x\left(x - 3\right)\left(x+3\right)\\[/latex]
53. [latex]f\left(x\right)=x\left(14 - 2x\right)\left(10 - 2x\right)\\[/latex]
54. [latex]f\left(x\right)=x\left(14 - 2x\right){\left(10 - 2x\right)}^{2}\\[/latex]
55. [latex]f\left(x\right)={x}^{3}-16x\\[/latex]
56. [latex]f\left(x\right)={x}^{3}-27\\[/latex]
57. [latex]f\left(x\right)={x}^{4}-81\\[/latex]
58. [latex]f\left(x\right)=-{x}^{3}+{x}^{2}+2x\\[/latex]
59. [latex]f\left(x\right)={x}^{3}-2{x}^{2}-15x\\[/latex]
60. [latex]f\left(x\right)={x}^{3}-0.01x\\[/latex]
For the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or –1. There may be more than one correct answer.
61. The y-intercept is [latex]\left(0,-4\right)\\[/latex]. The x-intercepts are [latex]\left(-2,0\right),\left(2,0\right)\\[/latex]. Degree is 2.
End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty \\[/latex].
62. The y-intercept is [latex]\left(0,9\right)\\[/latex]. The x-intercepts are [latex]\left(-3,0\right),\left(3,0\right)\\[/latex]. Degree is 2.
End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty\\ [/latex].
63. The y-intercept is [latex]\left(0,0\right)\\[/latex]. The x-intercepts are [latex]\left(0,0\right),\left(2,0\right)\\[/latex]. Degree is 3.
End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to -\infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty \\[/latex].
64. The y-intercept is [latex]\left(0,1\right)\\[/latex]. The x-intercept is [latex]\left(1,0\right)\\[/latex]. Degree is 3.
End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to -\infty \\[/latex].
65. The y-intercept is [latex]\left(0,1\right)\\[/latex]. There is no x-intercept. Degree is 4.
End behavior: [latex]\text{as }x\to -\infty ,f\left(x\right)\to \infty ,\text{ as }x\to \infty ,f\left(x\right)\to \infty\\[/latex].
For the following exercises, use the written statements to construct a polynomial function that represents the required information.
66. An oil slick is expanding as a circle. The radius of the circle is increasing at the rate of 20 meters per day. Express the area of the circle as a function of d, the number of days elapsed.
67. A cube has an edge of 3 feet. The edge is increasing at the rate of 2 feet per minute. Express the volume of the cube as a function of m, the number of minutes elapsed.
68. A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by x inches and the width increased by twice that amount, express the area of the rectangle as a function of x.
69. An open box is to be constructed by cutting out square corners of x-inch sides from a piece of cardboard 8 inches by 8 inches and then folding up the sides. Express the volume of the box as a function of x.
70. A rectangle is twice as long as it is wide. Squares of side 2 feet are cut out from each corner. Then the sides are folded up to make an open box. Express the volume of the box as a function of the width (x).Licenses & Attributions
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- Precalculus. Provided by: OpenStax Authored by: Jay Abramson, et al.. Located at: https://openstax.org/books/precalculus/pages/1-introduction-to-functions. License: CC BY: Attribution. License terms: Download For Free at : http://cnx.org/contents/[email protected]..
