{"id":1283,"date":"2015-11-12T18:35:30","date_gmt":"2015-11-12T18:35:30","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1283"},"modified":"2017-03-31T22:37:15","modified_gmt":"2017-03-31T22:37:15","slug":"section-exercises-48","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/section-exercises-48\/","title":{"raw":"Section Exercises","rendered":"Section Exercises"},"content":{"raw":"

1. Explain the difference between the coefficient of a power function and its degree.\r\n\r\n2.\u00a0If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?\r\n\r\n3. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.\r\n\r\n4.\u00a0What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?\r\n\r\n5. 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]\u00a0and as [latex]x\\to \\infty ,f\\left(x\\right)\\to -\\infty [\/latex].\r\n\r\nFor the following exercises, identify the function as a power function, a polynomial function, or neither.\r\n\r\n6. [latex]f\\left(x\\right)={x}^{5}[\/latex]\r\n\r\n7. [latex]f\\left(x\\right)={\\left({x}^{2}\\right)}^{3}[\/latex]\r\n\r\n8.\u00a0[latex]f\\left(x\\right)=x-{x}^{4}[\/latex]\r\n\r\n9. [latex]f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}-1}[\/latex]\r\n\r\n10.\u00a0[latex]f\\left(x\\right)=2x\\left(x+2\\right){\\left(x - 1\\right)}^{2}[\/latex]\r\n\r\n11. [latex]f\\left(x\\right)={3}^{x+1}[\/latex]\r\n\r\nFor the following exercises, find the degree and leading coefficient for the given polynomial.\r\n\r\n12. [latex]-3x{}^{4}[\/latex]\r\n\r\n13. [latex]7 - 2{x}^{2}[\/latex]\r\n\r\n14.\u00a0[latex]-2{x}^{2}- 3{x}^{5}+ x - 6 [\/latex]\r\n\r\n15. [latex]x\\left(4-{x}^{2}\\right)\\left(2x+1\\right)[\/latex]\r\n\r\n16.\u00a0[latex]{x}^{2}{\\left(2x - 3\\right)}^{2}[\/latex]\r\n\r\nFor the following exercises, determine the end behavior of the functions.\r\n\r\n17. [latex]f\\left(x\\right)={x}^{4}[\/latex]\r\n\r\n18.\u00a0[latex]f\\left(x\\right)={x}^{3}[\/latex]\r\n\r\n19. [latex]f\\left(x\\right)=-{x}^{4}[\/latex]\r\n\r\n20.\u00a0[latex]f\\left(x\\right)=-{x}^{9}[\/latex]\r\n\r\n21. [latex]f\\left(x\\right)=-2{x}^{4}- 3{x}^{2}+ x - 1[\/latex]\r\n\r\n22.\u00a0[latex]f\\left(x\\right)=3{x}^{2}+ x - 2[\/latex]\r\n\r\n23. [latex]f\\left(x\\right)={x}^{2}\\left(2{x}^{3}-x+1\\right)[\/latex]\r\n\r\n24. [latex]f\\left(x\\right)={\\left(2-x\\right)}^{7}[\/latex]\r\n\r\nFor the following exercises, find the intercepts of the functions.\r\n\r\n25. [latex]f\\left(t\\right)=2\\left(t - 1\\right)\\left(t+2\\right)\\left(t - 3\\right)[\/latex]\r\n\r\n26.\u00a0[latex]g\\left(n\\right)=-2\\left(3n - 1\\right)\\left(2n+1\\right)[\/latex]\r\n\r\n27. [latex]f\\left(x\\right)={x}^{4}-16[\/latex]\r\n\r\n28.\u00a0[latex]f\\left(x\\right)={x}^{3}+27[\/latex]\r\n\r\n29. [latex]f\\left(x\\right)=x\\left({x}^{2}-2x - 8\\right)[\/latex]\r\n\r\n30.\u00a0[latex]f\\left(x\\right)=\\left(x+3\\right)\\left(4{x}^{2}-1\\right)[\/latex]\r\n\r\nFor the following exercises, determine the least possible degree of the polynomial function shown.\r\n\r\n31.\r\n\"Graph\r\n\r\n32.\r\n\"Graph\r\n\r\n33.\r\n\"Graph\r\n\r\n34.\r\n\"Graph\r\n\r\n35.\r\n\"Graph\r\n\r\n36.\r\n\"Graph\r\n\r\n37.\r\n\"Graph\r\n\r\n38.\r\n\"Graph\r\n\r\nFor 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.\r\n\r\n39.\r\n\"Graph\r\n\r\n40.\r\n\"Graph\r\n\r\n41.\r\n\"Graph\r\n\r\n42.\r\n\"Graph\r\n\r\n43.\r\n\"Graph\r\n\r\n44.\r\n\"Graph\r\n\r\n45.\r\n\"Graph\r\n\r\nFor the following exercises, make a table to confirm the end behavior of the function.\r\n\r\n46. [latex]f\\left(x\\right)=-{x}^{3}[\/latex]\r\n\r\n47. [latex]f\\left(x\\right)={x}^{4}-5{x}^{2}[\/latex]\r\n\r\n48.\u00a0[latex]f\\left(x\\right)={x}^{2}{\\left(1-x\\right)}^{2}[\/latex]\r\n\r\n49. [latex]f\\left(x\\right)=\\left(x - 1\\right)\\left(x - 2\\right)\\left(3-x\\right)[\/latex]\r\n\r\n50.\u00a0[latex]f\\left(x\\right)=\\frac{{x}^{5}}{10}-{x}^{4}[\/latex]\r\n\r\nFor the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.\r\n\r\n51. [latex]f\\left(x\\right)={x}^{3}\\left(x - 2\\right)[\/latex]\r\n\r\n52. [latex]f\\left(x\\right)=x\\left(x - 3\\right)\\left(x+3\\right)[\/latex]\r\n\r\n53. [latex]f\\left(x\\right)=x\\left(14 - 2x\\right)\\left(10 - 2x\\right)[\/latex]\r\n\r\n54.\u00a0[latex]f\\left(x\\right)=x\\left(14 - 2x\\right){\\left(10 - 2x\\right)}^{2}[\/latex]\r\n\r\n55. [latex]f\\left(x\\right)={x}^{3}-16x[\/latex]\r\n\r\n56.\u00a0[latex]f\\left(x\\right)={x}^{3}-27[\/latex]\r\n\r\n57. [latex]f\\left(x\\right)={x}^{4}-81[\/latex]\r\n\r\n58.\u00a0[latex]f\\left(x\\right)=-{x}^{3}+{x}^{2}+2x[\/latex]\r\n\r\n59. [latex]f\\left(x\\right)={x}^{3}-2{x}^{2}-15x[\/latex]\r\n\r\n60. [latex]f\\left(x\\right)={x}^{3}-0.01x[\/latex]\r\n\r\nFor the following exercises, use the information about the graph of a polynomial function to determine the function. Assume the leading coefficient is 1 or \u20131. There may be more than one correct answer.\r\n\r\n61. The y<\/em>-intercept is [latex]\\left(0,-4\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-2,0\\right),\\left(2,0\\right)[\/latex]. Degree is 2.\r\n\r\nEnd 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].\r\n\r\n62.\u00a0The y<\/em>-intercept is [latex]\\left(0,9\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-3,0\\right),\\left(3,0\\right)[\/latex]. Degree is 2.\r\n\r\nEnd 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].\r\n\r\n63. The y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(2,0\\right)[\/latex]. Degree is 3.\r\n\r\nEnd 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].\r\n\r\n64.\u00a0The y-<\/em>intercept is [latex]\\left(0,1\\right)[\/latex]. The x<\/em>-intercept is [latex]\\left(1,0\\right)[\/latex]. Degree is 3.\r\n\r\nEnd 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].\r\n\r\n65. The y<\/em>-intercept is [latex]\\left(0,1\\right)[\/latex]. There is no x<\/em>-intercept. Degree is 4.\r\n\r\nEnd 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].\r\n\r\nFor the following exercises, use the written statements to construct a polynomial function that represents the required information.\r\n\r\n66. 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<\/em>, the number of days elapsed.\r\n\r\n67. 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<\/em>, the number of minutes elapsed.\r\n\r\n68.\u00a0A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by x<\/em>\u00a0inches and the width increased by twice that amount, express the area of the rectangle as a function of x<\/em>.\r\n\r\n69. An open box is to be constructed by cutting out square corners of x<\/em>-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<\/em>.\r\n\r\n70.\u00a0A 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<\/em>).<\/p>","rendered":"

1. Explain the difference between the coefficient of a power function and its degree.<\/p>\n

2.\u00a0If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?<\/p>\n

3. In general, explain the end behavior of a power function with odd degree if the leading coefficient is positive.<\/p>\n

4.\u00a0What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?<\/p>\n

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]\u00a0and as [latex]x\\to \\infty ,f\\left(x\\right)\\to -\\infty[\/latex].<\/p>\n

For the following exercises, identify the function as a power function, a polynomial function, or neither.<\/p>\n

6. [latex]f\\left(x\\right)={x}^{5}[\/latex]<\/p>\n

7. [latex]f\\left(x\\right)={\\left({x}^{2}\\right)}^{3}[\/latex]<\/p>\n

8.\u00a0[latex]f\\left(x\\right)=x-{x}^{4}[\/latex]<\/p>\n

9. [latex]f\\left(x\\right)=\\frac{{x}^{2}}{{x}^{2}-1}[\/latex]<\/p>\n

10.\u00a0[latex]f\\left(x\\right)=2x\\left(x+2\\right){\\left(x - 1\\right)}^{2}[\/latex]<\/p>\n

11. [latex]f\\left(x\\right)={3}^{x+1}[\/latex]<\/p>\n

For the following exercises, find the degree and leading coefficient for the given polynomial.<\/p>\n

12. [latex]-3x{}^{4}[\/latex]<\/p>\n

13. [latex]7 - 2{x}^{2}[\/latex]<\/p>\n

14.\u00a0[latex]-2{x}^{2}- 3{x}^{5}+ x - 6[\/latex]<\/p>\n

15. [latex]x\\left(4-{x}^{2}\\right)\\left(2x+1\\right)[\/latex]<\/p>\n

16.\u00a0[latex]{x}^{2}{\\left(2x - 3\\right)}^{2}[\/latex]<\/p>\n

For the following exercises, determine the end behavior of the functions.<\/p>\n

17. [latex]f\\left(x\\right)={x}^{4}[\/latex]<\/p>\n

18.\u00a0[latex]f\\left(x\\right)={x}^{3}[\/latex]<\/p>\n

19. [latex]f\\left(x\\right)=-{x}^{4}[\/latex]<\/p>\n

20.\u00a0[latex]f\\left(x\\right)=-{x}^{9}[\/latex]<\/p>\n

21. [latex]f\\left(x\\right)=-2{x}^{4}- 3{x}^{2}+ x - 1[\/latex]<\/p>\n

22.\u00a0[latex]f\\left(x\\right)=3{x}^{2}+ x - 2[\/latex]<\/p>\n

23. [latex]f\\left(x\\right)={x}^{2}\\left(2{x}^{3}-x+1\\right)[\/latex]<\/p>\n

24. [latex]f\\left(x\\right)={\\left(2-x\\right)}^{7}[\/latex]<\/p>\n

For the following exercises, find the intercepts of the functions.<\/p>\n

25. [latex]f\\left(t\\right)=2\\left(t - 1\\right)\\left(t+2\\right)\\left(t - 3\\right)[\/latex]<\/p>\n

26.\u00a0[latex]g\\left(n\\right)=-2\\left(3n - 1\\right)\\left(2n+1\\right)[\/latex]<\/p>\n

27. [latex]f\\left(x\\right)={x}^{4}-16[\/latex]<\/p>\n

28.\u00a0[latex]f\\left(x\\right)={x}^{3}+27[\/latex]<\/p>\n

29. [latex]f\\left(x\\right)=x\\left({x}^{2}-2x - 8\\right)[\/latex]<\/p>\n

30.\u00a0[latex]f\\left(x\\right)=\\left(x+3\\right)\\left(4{x}^{2}-1\\right)[\/latex]<\/p>\n

For the following exercises, determine the least possible degree of the polynomial function shown.<\/p>\n

31.
\n\"Graph<\/p>\n

32.
\n\"Graph<\/p>\n

33.
\n\"Graph<\/p>\n

34.
\n\"Graph<\/p>\n

35.
\n\"Graph<\/p>\n

36.
\n\"Graph<\/p>\n

37.
\n\"Graph<\/p>\n

38.
\n\"Graph<\/p>\n

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.<\/p>\n

39.
\n\"Graph<\/p>\n

40.
\n\"Graph<\/p>\n

41.
\n\"Graph<\/p>\n

42.
\n\"Graph<\/p>\n

43.
\n\"Graph<\/p>\n

44.
\n\"Graph<\/p>\n

45.
\n\"Graph<\/p>\n

For the following exercises, make a table to confirm the end behavior of the function.<\/p>\n

46. [latex]f\\left(x\\right)=-{x}^{3}[\/latex]<\/p>\n

47. [latex]f\\left(x\\right)={x}^{4}-5{x}^{2}[\/latex]<\/p>\n

48.\u00a0[latex]f\\left(x\\right)={x}^{2}{\\left(1-x\\right)}^{2}[\/latex]<\/p>\n

49. [latex]f\\left(x\\right)=\\left(x - 1\\right)\\left(x - 2\\right)\\left(3-x\\right)[\/latex]<\/p>\n

50.\u00a0[latex]f\\left(x\\right)=\\frac{{x}^{5}}{10}-{x}^{4}[\/latex]<\/p>\n

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.<\/p>\n

51. [latex]f\\left(x\\right)={x}^{3}\\left(x - 2\\right)[\/latex]<\/p>\n

52. [latex]f\\left(x\\right)=x\\left(x - 3\\right)\\left(x+3\\right)[\/latex]<\/p>\n

53. [latex]f\\left(x\\right)=x\\left(14 - 2x\\right)\\left(10 - 2x\\right)[\/latex]<\/p>\n

54.\u00a0[latex]f\\left(x\\right)=x\\left(14 - 2x\\right){\\left(10 - 2x\\right)}^{2}[\/latex]<\/p>\n

55. [latex]f\\left(x\\right)={x}^{3}-16x[\/latex]<\/p>\n

56.\u00a0[latex]f\\left(x\\right)={x}^{3}-27[\/latex]<\/p>\n

57. [latex]f\\left(x\\right)={x}^{4}-81[\/latex]<\/p>\n

58.\u00a0[latex]f\\left(x\\right)=-{x}^{3}+{x}^{2}+2x[\/latex]<\/p>\n

59. [latex]f\\left(x\\right)={x}^{3}-2{x}^{2}-15x[\/latex]<\/p>\n

60. [latex]f\\left(x\\right)={x}^{3}-0.01x[\/latex]<\/p>\n

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 \u20131. There may be more than one correct answer.<\/p>\n

61. The y<\/em>-intercept is [latex]\\left(0,-4\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-2,0\\right),\\left(2,0\\right)[\/latex]. Degree is 2.<\/p>\n

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].<\/p>\n

62.\u00a0The y<\/em>-intercept is [latex]\\left(0,9\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(-3,0\\right),\\left(3,0\\right)[\/latex]. Degree is 2.<\/p>\n

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].<\/p>\n

63. The y<\/em>-intercept is [latex]\\left(0,0\\right)[\/latex]. The x<\/em>-intercepts are [latex]\\left(0,0\\right),\\left(2,0\\right)[\/latex]. Degree is 3.<\/p>\n

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].<\/p>\n

64.\u00a0The y-<\/em>intercept is [latex]\\left(0,1\\right)[\/latex]. The x<\/em>-intercept is [latex]\\left(1,0\\right)[\/latex]. Degree is 3.<\/p>\n

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].<\/p>\n

65. The y<\/em>-intercept is [latex]\\left(0,1\\right)[\/latex]. There is no x<\/em>-intercept. Degree is 4.<\/p>\n

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].<\/p>\n

For the following exercises, use the written statements to construct a polynomial function that represents the required information.<\/p>\n

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<\/em>, the number of days elapsed.<\/p>\n

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<\/em>, the number of minutes elapsed.<\/p>\n

68.\u00a0A rectangle has a length of 10 inches and a width of 6 inches. If the length is increased by x<\/em>\u00a0inches and the width increased by twice that amount, express the area of the rectangle as a function of x<\/em>.<\/p>\n

69. An open box is to be constructed by cutting out square corners of x<\/em>-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<\/em>.<\/p>\n

70.\u00a0A 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<\/em>).<\/p>\n\n\t\t\t

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