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3.4e: Exercises - Polynomial Graphs

( \newcommand{\kernel}{\mathrm{null}\,}\)

A: Concepts 

Exercise 3.4e.A

1) What is the difference between an x-intercept and a zero of a polynomial function f?

2) If a polynomial function of degree n has n distinct zeros, what do you know about the graph of the function?

3) What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph? .

4) Explain how the factored form of the polynomial helps us in graphing it.

5) If the graph of a polynomial just touches the x-axis and then changes direction, what can we conclude about the factored form of the polynomial?

Answers to odd exercises:

1. The x-intercept is where the graph of the function crosses the x-axis, and the zero of the function is the input value for which f(x)=0.

3. The maximum number of turning points is one less than the degree of the polynomial.

5. There will be a factor raised to an even power.

B: Multiplicity from an Equation

Exercise 3.4e.B

 Find the zeros and give the multiplicity of each.

6) f(x)=(x+2)3(x3)2

7) f(x)=x2(2x+3)5(x4)2

8) f(x)=x3(x1)3(x+2)

9) f(x)=x2(x2+4x+4)

10) f(x)=(2x+1)3(9x26x+1)

11) f(x)=(3x+2)5(x210x+25)

12) f(x)=x(4x212x+9)(x2+8x+16)

13) f(x)=x6x52x4

14) f(x)=3x4+6x3+3x2

15) f(x)=4x512x4+9x3

16) f(x)=2x4(x34x2+4x)

17) f(x)=4x4(9x412x3+4x2)

Answers to odd exercises:

7. 0 and 4 with multiplicity 2,  32 with multiplicity 5

9. 0 with multiplicity 22 with multiplicity 2

11. 23 with multiplicity 5, 5 with multiplicity 2

13. 0 with multiplicity 4,  2 and 1 with multiplicity 1

15. 32 with multiplicity 2,  0 with multiplicity 3

17. 0 with multiplicity 623 with multiplicity 2

C: Multiplicity from a Graph

Exercise 3.4e.C

 Use the graph to identify zeros and multiplicity.

19)

CNX_PreCalc_Figure_03_04_212.jpg

20)

CNX_PreCalc_Figure_03_04_213.jpg

21)

CNX_PreCalc_Figure_03_04_214.jpg

22)

CNX_PreCalc_Figure_03_04_215.jpg

Answers to odd exercises:

19. 4,2,1,3 with multiplicity 1

21. 2,3 each with multiplicity 2

D: Graph polynomials

Exercise 3.4e.D 

 Graph the polynomial functions. State the x- and y- intercepts, multiplicity, and end behavior.

24) f(x)=(x+3)2(x2)

25) g(x)=(x+4)(x1)2

26) h(x)=(x1)3(x+3)2

27) k(x)=(x3)3(x2)2

28) m(x)=2x(x1)(x+3)

29) n(x)=3x(x+2)(x4)

30. a(x)=x(x+2)2

31. g(x)=x(x+2)3

32. f(x)=2(x2)2(x+1)

33. g(x)=(2x+1)2(x3)

34. f(x)=x3(x+2)2

35. P(x)=(x1)(x2)(x3)(x4)

36. q(x)=(x+5)2(x3)4

37. h(x)=x2(x2)2(x+2)2

38. h(t)=(3t)(t2+1)

39. Z(b)=b(42b2)

Answers to odd exercises:

25. x-intercepts, (1,0) with multiplicity 2, (4,0) with multiplicity 1, y- intercept (0,4) . As x, f(x), as x, f(x).

CNX_Precalc_Figure_03_04_202.jpg

27. x-intercepts (3,0) with multiplicity 3, (2,0) with multiplicity 2, y- intercept (0,108). As x, f(x), as x, f(x).

3.4 example 27.png

29. x-intercepts (0,0),(2,0),(4,0) with multiplicity 1, y-intercept (0, 0). As x, f(x), as x, f(x).

CNX_Precalc_Figure_03_04_206.jpg

31. (2,0) multiplicity 3,
      (0,0) multiplicity 1, 
      y-intercept (0,0),
      end behaviour: 
      3.4 example 31.png
33. (12,0) multiplicity 2,
      (3,0) multiplicity 1
      y-intercept (0,3),
      end behaviour: 
     3.4 example 33.png
35. (1,0)(2,0), (3,0), (4,0)
      all multiplicity 1,
      y-intercept  (0,24),
      end behaviour: 
     3.4 example 35.png
37. (2,0), (2,0)(0,0) 
      all multiplicity 2, y-intercept (0,0),
      end behaviour: 
    3.4 example 37.png
39. (42,0), (42,0), (0,0)
       all multiplicity 1, y-intercept (0,0),
       end behaviour: 
     3.4 example 39.png
 

 Graph the polynomial functions. State the x- and y- intercepts, multiplicity, and end behavior.

41. f(x)=(x+3)2(x2)

42. g(x)=(x+4)(x1)2

43. h(x)=(x1)3(x+3)2

44. k(x)=(x3)3(x2)2 

45. m(x)=2x(x1)(x+3)

46. n(x)=3x(x+2)(x4)

47. f(x)=9xx3

48. f(x)=8+x3

49. f(x)=x425x2

50. f(x)=16x4

51. f(x)=x4+2x3+8x2

52. f(x)=x3+7x29x

53. f(x)=2x3+12x28x48

54. f(x)=4x4+10x34x210x

Answers to odd exercises:
41. (3,0) multiplicity 2,
      (2,0) multiplicity 1
      y-intercept(0,18),  
Screen Shot 2019-10-03 at 6.05.35 PM.png
43. (3,0) multiplicity 2,
      (1,0) multiplicity 3,
      y-intercept(0,9) Screen Shot 2019-10-03 at 6.05.54 PM.png
45. (3,0)(0,0)(1,0)
      all multiplicity 1
      y-intercept(0,0)
Screen Shot 2019-10-03 at 6.06.33 PM.png
47. (3,0)(0,0)(3,0)
      all multiplicity 1
      y-intercept (0,0)
3.4 example 47.png
49. (5,0)(5,0) both multiplicity 1,
      (0,0) multiplicity 2
      y-intercept (0,0)
3.4 example 49.png
51. (2,0)(4,0) both multiplicity 1
      (0,0) multiplicity 2
      y-intercept (0,0)
3.4 example 51.png
53. (6,0)(2,0)(2,0)
      all multiplicity 1
      y-intercept (0,48)
3.4 example 53.png
   

E: Polynomial Degree from a Graph

Exercise 3.4e.E

 Determine the least possible degree of the polynomial function shown.

61)

CNX_Precalc_Figure_03_03_201.jpg

62)

CNX_Precalc_Figure_03_03_202.jpg

63)

CNX_Precalc_Figure_03_03_203.jpg

64)

CNX_Precalc_Figure_03_03_204.jpg

65)

CNX_Precalc_Figure_03_03_205.jpg

66)

CNX_Precalc_Figure_03_03_206.jpg

67)

CNX_Precalc_Figure_03_03_207.jpg

68)

CNX_Precalc_Figure_03_03_208.jpg

Answers to odd exercises:

61. 3 63. 5 65. 3 67. 5

F: Construct an Equation from a graph

Exercise 3.4e.F 

 Use the graphs to write the formula for the polynomial function of least degree.

69)

CNX_PreCalc_Figure_03_04_208.jpg

70)

CNX_Precalc_Figure_03_04_207.jpg

71)

CNX_PreCalc_Figure_03_04_210.jpg

72)

CNX_PreCalc_Figure_03_04_209.jpg

73.

屏幕快照 2019-06-23 上午3.25.44.png

74)

CNX_PreCalc_Figure_03_04_211.jpg
Answers to odd exercises:

69. f(x)=(x+3)(x+1)(x3)   or  f(x)=29(x+3)(x+1)(x3)
71. f(x)=(x+2)2(x3)   or  f(x)=14(x+2)2(x3)
73. f(x)=(x+3)(x+2)(x2)(x4)   or  f(x)=124(x+3)(x+2)(x2)(x4)

 Use the graphs to write a formula for the polynomial function of least degree.

75)

CNX_PreCalc_Figure_03_04_217.jpg 

76)

CNX_PreCalc_Figure_03_04_216.jpg

77)

CNX_PreCalc_Figure_03_04_218.jpg 

78)

CNX_Precalc_Figure_03_03_216.jpg

79)

CNX_Precalc_Figure_03_03_222.jpg 

80)

CNX_Precalc_Figure_03_03_220.jpg

81)

CNX_Precalc_Figure_03_03_218.jpg 

82)

CNX_Precalc_Figure_03_03_224.jpg

Answers to odd exercises:

75. f(x)=(x500)2(x+200)    77. f(x)=(x+300)2(x100)3   
79. f(x)=(x+3)(x3)(x2+10)    81. f(x)=4x(x5)(x7)     

 Use the graphs to write a formula for the polynomial function of least degree.

83.

屏幕快照 2019-06-23 上午3.26.29.png

84(a).

屏幕快照 2019-06-23 上午3.26.12.png

84(b).

屏幕快照 2019-06-23 上午3.26.46.png

85.

屏幕快照 2019-06-23 上午5.32.03.png

86.

屏幕快照 2019-06-23 上午5.32.31.png

 

87.

屏幕快照 2019-06-23 上午5.32.47.png

88.

屏幕快照 2019-06-23 上午5.33.17.png

89.

.屏幕快照 2019-06-23 上午5.33.39.png

90.

屏幕快照 2019-06-23 上午5.34.00.png

Answers to odd exercises:

83. y=124(x+4)(x+2)(x3)2

85. y=112(x+2)2(x3)2

87. y=16(x+3)(x+2)(x1)3

89. y=116(x+3)(x+1)(x2)2(x4)

G: Construct an Equation from a Description

Exercise 3.4e.G 

 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.

91) The y-intercept is (0,4). The x-intercepts are (2,0),(2,0). Degree is 2. End behavior:  

92) The y-intercept is (0,9). The x-intercepts are (3,0),(3,0). Degree is 2. End behavior:  

93) The y-intercept is (0,0). The x-intercepts are (0,0),(2,0). Degree is 3. End behavior:  

94) The y-intercept is (0,1). The x-intercept is (1,0). Degree is 3. End behavior:  

95) The y-intercept is (0,1). There is no x-intercept. Degree is 4. End behavior:  

 Use the given information about the polynomial graph to write the equation.

97) Degree 3. Zeros at x=2,x=1, and x=3. y-intercept at (0,4)

98) Degree 3. Zeros at x=5, x=2, and x=1. y-intercept at (0,6)

99) Degree 5. Roots of multiplicity 2 at x=3 and x=1. Root of multiplicity 1 at x=3.
y-intercept at (0,9)

100) Degree 4. Root of multiplicity 2 at x=4. Roots of multiplicity 1 at x=1 and x=2.
y-intercept at (0,3)

101) Degree 5. Double zero at x=1. Triple zero at x=3. Passes through the point (2,15)

102) Degree 3. Zeros at x=4, x=3, and x=2. y-intercept at (0,24)

103) Degree 3. Zeros at x=3, x=2 and x=1. y-intercept at (0,12)

104) Degree 5. Roots of multiplicity 2 at x=3 and x=2. Root of multiplicity 1 at x=2. y-intercept at (0,4).

105) Degree 4. Roots of multiplicity 2 at x=12. Roots of multiplicity 1 at x=6 and x=2. y-intercept at (0,18)

106) Double zero at x=3. Triple zero at x=0. Passes through the point (1,32).

107. Degree 3. Zeros at x = -2, x = 1, and x = 3. Vertical intercept at (0, -4)

108. Degree 3. Zeros at x = -5, x = -2, and x = 1. Vertical intercept at (0, 6)

109. Degree 5. Roots of multiplicity 2 at x = 3 and x = 1. Root of multiplicity 1 at x = -3. Vertical intercept at (0, 9)

110. Degree 4. Root of multiplicity 2 at x = 4. Roots of multiplicity 1 at x = 1 and x = -2.
Vertical intercept at (0, -3)

111. Degree 5. Double zero at x = 1. Triple zero at x = 3. Passes through the point (2, 15)

112. Degree 5. Single zero at x = -2 and x = 3. Triple zero at x = 1. Passes through the point (2, 4)

Answers to odd exercises:

91. f(x)=x24

93. f(x)=x34x2+4x

95. f(x)=x4+1

97. f(x)=23(x+2)(x1)(x3)

99. f(x)=13(x3)2(x1)2(x+3)

101. f(x)=15(x1)2(x3)3

103. (f(x)=−2(x+3)(x+2)(x−1)\)

105. f(x)=32(2x1)2(x6)(x+2)

107. y=23(x+2)(x1)(x3)

109. y=13(x1)2(x3)2(x+3)

111. y=15(x1)2(x3)3

H: Turning Points

Exercise 3.4e.H

 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.

115)

CNX_Precalc_Figure_03_03_209.jpg

116)

CNX_Precalc_Figure_03_03_210.jpg

117)

CNX_Precalc_Figure_03_03_211.jpg

118)

CNX_Precalc_Figure_03_03_212.jpg

119)

CNX_Precalc_Figure_03_03_213.jpg

120)

CNX_Precalc_Figure_03_03_214.jpg

121)

CNX_Precalc_Figure_03_03_215.jpg

 

Answers to odd exercises:

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

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

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

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


3.4e: Exercises - Polynomial Graphs is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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