Problem Set 14: Power Functions and Polynomial Functions

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 x,f(x) and as x,f(x).

For the following exercises, identify the function as a power function, a polynomial function, or neither.

6. f(x)=x5

7. f(x)=(x2)3

8. f(x)=xx4

9. f(x)=x2x21

10. f(x)=2x(x+2)(x1)2

11. f(x)=3x+1

For the following exercises, find the degree and leading coefficient for the given polynomial.

12. 3x4

13. 72x2

14. 2x23x5+x6

15. x(4x2)(2x+1)

16. x2(2x3)2

For the following exercises, determine the end behavior of the functions.

17. f(x)=x4

18. f(x)=x3

19. f(x)=x4

20. f(x)=x9

21. f(x)=2x43x2+x1

22. f(x)=3x2+x2

23. f(x)=x2(2x3x+1)

24. f(x)=(2x)7

For the following exercises, find the intercepts of the functions.

25. f(t)=2(t1)(t+2)(t3)

26. g(n)=2(3n1)(2n+1)

27. f(x)=x416

28. f(x)=x3+27

29. f(x)=x(x22x8)

30. f(x)=(x+3)(4x21)

For the following exercises, determine the least possible degree of the polynomial function shown.

31.
Graph of an odd-degree polynomial.

32.
Graph of an even-degree polynomial.

33.
Graph of an odd-degree polynomial.

34.
Graph of an odd-degree polynomial.

35.
Graph of an odd-degree polynomial.

36.
Graph of an even-degree polynomial.

37.
Graph of an odd-degree polynomial.

38.
Graph of an even-degree polynomial.

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.

39.
Graph of an odd-degree polynomial.

40.
Graph of an equation.

41.
Graph of an even-degree polynomial.

42.
Graph of an odd-degree polynomial.

43.
Graph of an odd-degree polynomial.

44.
Graph of an equation.

45.
Graph of an odd-degree polynomial.

For the following exercises, make a table to confirm the end behavior of the function.

46. f(x)=x3

47. f(x)=x45x2

48. f(x)=x2(1x)2

49. f(x)=(x1)(x2)(3x)

50. f(x)=x510x4

For the following exercises, graph the polynomial functions using a calculator. Based on the graph, determine the intercepts and the end behavior.

51. f(x)=x3(x2)

52. f(x)=x(x3)(x+3)

53. f(x)=x(142x)(102x)

54. f(x)=x(142x)(102x)2

55. f(x)=x316x

56. f(x)=x327

57. f(x)=x481

58. f(x)=x3+x2+2x

59. f(x)=x32x215x

60. f(x)=x30.01x

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 (0,4). The x-intercepts are (2,0),(2,0). Degree is 2.

End behavior: as x,f(x), as x,f(x).

62. The y-intercept is (0,9). The x-intercepts are (3,0),(3,0). Degree is 2.

End behavior: as x,f(x), as x,f(x).

63. The y-intercept is (0,0). The x-intercepts are (0,0),(2,0). Degree is 3.

End behavior: as x,f(x), as x,f(x).

64. The y-intercept is (0,1). The x-intercept is (1,0). Degree is 3.

End behavior: as x,f(x), as x,f(x).

65. The y-intercept is (0,1). There is no x-intercept. Degree is 4.

End behavior: as x,f(x), as x,f(x).

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).