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Mathematics LibreTexts

3.3E: Exercises

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Section Exercises

Verbal

1. Explain the difference between the coefficient of a power function and its degree.

Answer: The coefficient of the power function is the real number that is multiplied by the variable raised to a power. The degree is the highest power appearing in the function.

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.

Answer: As x decreases without bound, so does f(x). As x increases without bound, so does f(x).

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).

Answer: The polynomial function is of even degree and leading coefficient is negative.

Algebraic

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

Answer: Power function

8. f(x)=xx4

Exercise 3.3E.9

f(x)=x2x21

Answer

Neither

10. f(x)=2x(x+2)(x1)2

11. f(x)=3x+1

Answer: Neither

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


12. 3x4

13. 72x2

Answer: Degree = 2, Coefficient = –2

14. 2x23x5+x6

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

Answer: Degree =4, Coefficient = –2

16. x2(2x3)2

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


17. f(x)=x4

Answer: As x, f(x), as x, f(x)

18. f(x)=x3

19. f(x)=x4

Answer: As x, f(x), as x, f(x)

20. f(x)=x9

21. f(x)=2x43x2+x1

Answer: As x, f(x), as x, f(x)

22. f(x)=3x2+x2

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

Answer: As x, f(x), as x, f(x)

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

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


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

Answer: y-intercept is (0,12), t-intercepts are (1,0);(2,0); and (3,0).

26. g(n)=2(3n1)(2n+1)

Exercise 3.3E.27

f(x)=x416

Answer

y-intercept is (0,16). x-intercepts are (2,0) and (2,0).

28. f(x)=x3+27

29. f(x)=x(x22x8)

Answer: y-intercept is (0,0). x-intercepts are (0,0),(4,0), and (2,0).

30. f(x)=(x+3)(4x21)

Graphical

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

31.

CNX_Precalc_Figure_03_03_201.jpg

Answer: 3

32.

CNX_Precalc_Figure_03_03_202.jpg

33.
CNX_Precalc_Figure_03_03_203.jpg

Answer: 5

34.

CNX_Precalc_Figure_03_03_204.jpg

35.
CNX_Precalc_Figure_03_03_205.jpg

Answer: 3

36.

CNX_Precalc_Figure_03_03_206.jpg

37.
CNX_Precalc_Figure_03_03_207.jpg

Answer: 5

38.

CNX_Precalc_Figure_03_03_208.jpg

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.

CNX_Precalc_Figure_03_03_209.jpg

Answer: Yes. Number of turning points is 2. Least possible degree is 3.

40.

CNX_Precalc_Figure_03_03_210.jpg

41.
CNX_Precalc_Figure_03_03_211.jpg

Answer: Yes. Number of turning points is 1. Least possible degree is 2.

42.

CNX_Precalc_Figure_03_03_212.jpg

43.
CNX_Precalc_Figure_03_03_213.jpg

Answer: Yes. Number of turning points is 0. Least possible degree is 3.

Exercise 3.3E.44

CNX_Precalc_Figure_03_03_214.jpg

Answer

No (the graph is not smooth)

45.

CNX_Precalc_Figure_03_03_215.jpg

Answer: Yes. Number of turning points is 0. Least possible degree is 1.

Numeric

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


46. f(x)=x3

47. f(x)=x45x2

Answer:

x f(x)
10 9,500
100 99,950,000
–10 9,500
–100 99,950,000

as x, f(x), as x, f(x)

48. f(x)=x2(1x)2

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

Answer:

x f(x)
10 9,500
100 99,950,000
–10 9,500
–100 99,950,000

as x, f(x), as x, f(x)

50. f(x)=x510x4

Technology

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)

Answer:

CNX_Precalc_Figure_03_03_216.jpg

The y-intercept is (0,0). The x-intercepts are (0,0),(2,0). As x, f(x), as x, f(x)

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

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

Answer:

CNX_Precalc_Figure_03_03_218.jpg

The y-intercept is (0,0) . The x-intercepts are (0,0),(5,0),(7,0). As x, f(x), as x, f(x)

54. f(x)=x(142x)(102x)2

55. f(x)=x316x

Answer:

CNX_Precalc_Figure_03_03_220.jpg

The y-intercept is (0, 0). The x-intercept is (4,0),(0,0),(4,0). As x, f(x), as x, f(x)

56. f(x)=x327

57. f(x)=x481

Answer:

CNX_Precalc_Figure_03_03_222.jpg

The y-intercept is (0, −81). The x-intercept are (3,0),(3,0). As x, f(x), as x, f(x)

58. f(x)=x3+x2+2x

59. f(x)=x32x215x

Answer:

CNX_Precalc_Figure_03_03_224.jpg

The y-intercept is (0,0). The x-intercepts are (3,0),(0,0),(5,0). As x, f(x), as x, f(x)

60. f(x)=x30.01x

Extensions

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).

Answer: f(x)=x24

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).

Answer: f(x)=x34x2+4x

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).

Answer: f(x)=x4+1

Real-World Applications

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.

Answer: V(m)=8m3+36m2+54m+27

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.

Exercise 3.3E.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.

Answer

V(x)=4x332x2+64x

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).


3.3E: Exercises is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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