1. Slide 2- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 2 Polynomial, Power, and Rational Functions

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.1 Linear and Quadratic Functions and Modeling

4. Slide 2- 4 Quick Review

5. Slide 2- 5 Quick Review Solutions

6. Slide 2- 6 What you’ll learn about Polynomial Functions Linear Functions and Their Graphs Average Rate of Change Linear Correlation and Modeling Quadratic Functions and Their Graphs Applications of Quadratic Functions … and why Many business and economic problems are modeled by linear functions. Quadratic and higher degree polynomial functions are used to model some manufacturing applications.

7. Slide 2- 7 Polynomial Function

8. Slide 2- 8 Polynomial Functions of No and Low Degree NameFormDegree Zero Functionf(x) = 0Undefined Constant Functionf(x) = a (a ≠ 0)0 Linear Functionf(x)=ax + b (a ≠ 0)1 Quadratic Functionf(x)=ax 2 + bx + c (a ≠ 0)2

9. Slide 2- 9 Example Finding an Equation of a Linear Function

10. Slide 2- 10 Example Finding an Equation of a Linear Function

11. Slide 2- 11 Average Rate of Change

12. Slide 2- 12 Constant Rate of Change Theorem A function defined on all real numbers is a linear function if and only if it has a constant nonzero average rate of change between any two points on its graph.

13. Slide 2- 13 Characterizing the Nature of a Linear Function Point of ViewCharacterization Verbalpolynomial of degree 1 Algebraic f(x) = mx + b (m ≠ 0) Graphicalslant line with slope m and y-intercept b Analyticalfunction with constant nonzero rate of change m: f is increasing if m > 0, decreasing if m < 0; initial value of the function = f(0) = b

14. Slide 2- 14 Properties of the Correlation Coefficient, r 1. -1 ≤ r ≤ 1 2. When r > 0, there is a positive linear correlation. 3. When r < 0, there is a negative linear correlation. 4. When |r| ≈ 1, there is a strong linear correlation. 5. When |r| ≈ 0, there is weak or no linear correlation.

15. Slide 2- 15 Linear Correlation

16. Slide 2- 16 Regression Analysis 1. Enter and plot the data (scatter plot). 2. Find the regression model that fits the problem situation. 3. Superimpose the graph of the regression model on the scatter plot, and observe the fit. 4. Use the regression model to make the predictions called for in the problem.

17. Slide 2- 17 Example Transforming the Squaring Function

18. Slide 2- 18 Example Transforming the Squaring Function

19. Slide 2- 19 The Graph of f(x)=ax 2

20. Slide 2- 20 Vertex Form of a Quadratic Equation Any quadratic function f(x) = ax 2 + bx + c, a ≠ 0, can be written in the vertex form f(x) = a(x – h) 2 + k The graph of f is a parabola with vertex (h,k) and axis x = h, where h = -b/(2a) and k = c – ah 2. If a > 0, the parabola opens upward, and if a < 0, it opens downward.

21. Slide 2- 21 Find the vertex and the line of symmetry of the graph y = (x – 1) 2 + 2 Domain Range (- ,  ) [2,  ) Vertex (1,2) x = 1

22. Slide 2- 22 Find the vertex and the line of symmetry of the graph y = -(x + 2) 2 - 3 Domain Range (- ,  ) (- ,-3] Vertex (-2,-3) x = -2

23. Slide 2- 23 Let f(x) = x 2 + 2x + 4. (a) Write f in standard form. (b) Determine the vertex of f. (c) Is the vertex a maximum or a minimum? Explain f(x) = x 2 + 2x + 4 f(x) = (x + 1) 2 + 3 Vertex (-1,3) opens up (-1,3) is a minimum + 1- 1

24. Slide 2- 24 Let f(x) = 2x 2 + 6x - 8. (a) Write f in standard form. (b) Determine the vertex of f. (c) Is the vertex a maximum or a minimum? Explain f(x) = 2(x + 3/2) 2 - 25/2 Vertex (-3/2,-25/2) opens up (-3/2,25/2) is a minimum + 9/4 - 9/2f(x) = 2(x 2 + 3x ) - 8

25. Slide 2- 25 If we perform completing the square process on f(x) = ax 2 + bx + c and write it in standard form, we get

26. Slide 2- 26 So the vertex is

27. Slide 2- 27 To get the coordinates of the vertex of any quadratic function, simply use the vertex formula. If a > 0, the parabola open up and the vertex is a minimum. If a < 0, the parabola opens down and the parabola is a maximum. 1.5 Quadratic Functions

28. Slide 2- 28 Example Finding the Vertex and Axis of a Quadratic Function

29. Slide 2- 29 Example Finding the Vertex and Axis of a Quadratic Function

30. Slide 2- 30 Characterizing the Nature of a Quadratic Function Point of View Characterization

31. Slide 2- 31 Vertical Free-Fall Motion

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.2 Power Functions and Modeling

33. Slide 2- 33 Quick Review

34. Slide 2- 34 Quick Review Solutions

35. Slide 2- 35 What you’ll learn about Power Functions and Variation Monomial Functions and Their Graphs Graphs of Power Functions Modeling with Power Functions … and why Power functions specify the proportional relationships of geometry, chemistry, and physics.

36. Slide 2- 36 Power Function Any function that can be written in the form f(x) = k·x a, where k and a are nonzero constants, is a power function. The constant a is the power, and the k is the constant of variation, or constant of proportion. We say f(x) varies as the a th power of x, or f(x) is proportional to the a th power of x.

37. Slide 2- 37 Example Analyzing Power Functions

38. Slide 2- 38 Example Analyzing Power Functions

39. Slide 2- 39 Monomial Function Any function that can be written as f(x) = k or f(x) = k·x n, where k is a constant and n is a positive integer, is a monomial function.

40. Slide 2- 40 Example Graphing Monomial Functions

41. Slide 2- 41 Example Graphing Monomial Functions

42. Slide 2- 42 Graphs of Power Functions For any power function f(x) = k·x a, one of the following three things happens when x < 0. f is undefined for x < 0. f is an even function. f is an odd function.

43. Slide 2- 43 Graphs of Power Functions

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.3 Polynomial Functions of Higher Degree with Modeling

45. Slide 2- 45 Quick Review

46. Slide 2- 46 Quick Review Solutions

47. Slide 2- 47 What you’ll learn about Graphs of Polynomial Functions End Behavior of Polynomial Functions Zeros of Polynomial Functions Intermediate Value Theorem Modeling … and why These topics are important in modeling and can be used to provide approximations to more complicated functions, as you will see if you study calculus.

48. Slide 2- 48 The Vocabulary of Polynomials

49. Slide 2- 49 Example Graphing Transformations of Monomial Functions

50. Slide 2- 50 Example Graphing Transformations of Monomial Functions

51. Slide 2- 51 Cubic Functions

52. Slide 2- 52 Quartic Function

53. Slide 2- 53 Local Extrema and Zeros of Polynomial Functions A polynomial function of degree n has at most n – 1 local extrema and at most n zeros.

54. Slide 2- 54 Leading Term Test for Polynomial End Behavior

55. Slide 2- 55 Example Applying Polynomial Theory

56. Slide 2- 56 Example Applying Polynomial Theory

57. Slide 2- 57 Example Finding the Zeros of a Polynomial Function

58. Slide 2- 58 Example Finding the Zeros of a Polynomial Function

59. Slide 2- 59 Multiplicity of a Zero of a Polynomial Function

60. Slide 2- 60 Example Sketching the Graph of a Factored Polynomial

61. Slide 2- 61 Example Sketching the Graph of a Factored Polynomial

62. Slide 2- 62 Intermediate Value Theorem If a and b are real numbers with a < b and if f is continuous on the interval [a,b], then f takes on every value between f(a) and f(b). In other words, if y 0 is between f(a) and f(b), then y 0 =f(c) for some number c in [a,b].

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.4 Real Zeros of Polynomial Functions

64. Slide 2- 64 Quick Review

65. Slide 2- 65 Quick Review Solutions

66. Slide 2- 66 What you’ll learn about Long Division and the Division Algorithm Remainder and Factor Theorems Synthetic Division Rational Zeros Theorem Upper and Lower Bounds … and why These topics help identify and locate the real zeros of polynomial functions.

67. Slide 2- 67 Division Algorithm for Polynomials

68. Slide 2- 68 Example Using Polynomial Long Division

69. Slide 2- 69 Example Using Polynomial Long Division

70. Slide 2- 70 Remainder Theorem

71. Slide 2- 71 Example Using the Remainder Theorem

72. Slide 2- 72 Example Using the Remainder Theorem

73. Slide 2- 73 Factor Theorem

74. Slide 2- 74 Example Using Synthetic Division

75. Slide 2- 75 Example Using Synthetic Division

76. Slide 2- 76 Rational Zeros Theorem

77. Slide 2- 77 Upper and Lower Bound Tests for Real Zeros

78. Slide 2- 78 Example Finding the Real Zeros of a Polynomial Function

79. Slide 2- 79 Example Finding the Real Zeros of a Polynomial Function

80. Slide 2- 80 Example Finding the Real Zeros of a Polynomial Function

81. Slide 2- 81 Example Finding the Real Zeros of a Polynomial Function

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.5 Complex Zeros and the Fundamental Theorem of Algebra

83. Slide 2- 83 Quick Review

84. Slide 2- 84 Quick Review Solutions

85. Slide 2- 85 What you’ll learn about Two Major Theorems Complex Conjugate Zeros Factoring with Real Number Coefficients … and why These topics provide the complete story about the zeros and factors of polynomials with real number coefficients.

86. Slide 2- 86 Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

87. Slide 2- 87 Linear Factorization Theorem

88. Slide 2- 88 Fundamental Polynomial Connections in the Complex Case The following statements about a polynomial function f are equivalent if k is a complex number: 1. x = k is a solution (or root) of the equation f(x) = 0 2. k is a zero of the function f. 3. x – k is a factor of f(x).

89. Slide 2- 89 Example Exploring Fundamental Polynomial Connections

90. Slide 2- 90 Example Exploring Fundamental Polynomial Connections

91. Slide 2- 91 Complex Conjugate Zeros

92. Slide 2- 92 Example Finding a Polynomial from Given Zeros

93. Slide 2- 93 Example Finding a Polynomial from Given Zeros

94. Slide 2- 94 Factors of a Polynomial with Real Coefficients Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors, each with real coefficients.

95. Slide 2- 95 Example Factoring a Polynomial

96. Slide 2- 96 Example Factoring a Polynomial

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.6 Graphs of Rational Functions

98. Slide 2- 98 Quick Review

99. Slide 2- 99 Quick Review Solutions

100. Slide 2- 100 What you’ll learn about Rational Functions Transformations of the Reciprocal Function Limits and Asymptotes Analyzing Graphs of Rational Functions … and why Rational functions are used in calculus and in scientific applications such as inverse proportions.

101. Slide 2- 101 Rational Functions

102. Slide 2- 102 Example Finding the Domain of a Rational Function

103. Slide 2- 103 Example Finding the Domain of a Rational Function

104. Slide 2- 104 Graph a Rational Function

105. Slide 2- 105 Graph a Rational Function

106. Slide 2- 106 Example Finding Asymptotes of Rational Functions

107. Slide 2- 107 Example Finding Asymptotes of Rational Functions

108. Slide 2- 108 Example Graphing a Rational Function

109. Slide 2- 109 Example Graphing a Rational Function

110. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.7 Solving Equations in One Variable

111. Slide 2- 111 Quick Review

112. Slide 2- 112 Quick Review Solutions

113. Slide 2- 113 What you’ll learn about Solving Rational Equations Extraneous Solutions Applications … and why Applications involving rational functions as models often require that an equation involving fractions be solved.

114. Slide 2- 114 Extraneous Solutions When we multiply or divide an equation by an expression containing variables, the resulting equation may have solutions that are not solutions of the original equation. These are extraneous solutions. For this reason we must check each solution of the resulting equation in the original equation.

115. Slide 2- 115 Example Solving by Clearing Fractions

116. Slide 2- 116 Example Solving by Clearing Fractions

117. Slide 2- 117 Example Eliminating Extraneous Solutions

118. Slide 2- 118 Example Eliminating Extraneous Solutions

119. Slide 2- 119 Example Finding a Minimum Perimeter

120. Slide 2- 120 Example Finding a Minimum Perimeter

121. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2.8 Solving Inequalities in One Variable

122. Slide 2- 122 Quick Review

123. Slide 2- 123 Quick Review Solutions

124. Slide 2- 124 What you’ll learn about Polynomial Inequalities Rational Inequalities Other Inequalities Applications … and why Designing containers as well as other types of applications often require that an inequality be solved.

125. Slide 2- 125 Polynomial Inequalities

126. Slide 2- 126 Example Finding where a Polynomial is Zero, Positive, or Negative

127. Slide 2- 127 Example Finding where a Polynomial is Zero, Positive, or Negative -34 (-)(-) 2 (+)(-) 2 (+)(+) 2 negative positive

128. Slide 2- 128 Example Solving a Polynomial Inequality Graphically

129. Slide 2- 129 Example Solving a Polynomial Inequality Graphically

130. Slide 2- 130 Example Creating a Sign Chart for a Rational Function

131. Slide 2- 131 Example Creating a Sign Chart for a Rational Function -31 (-) (-)(-) negative positive (-) (+)(-) (+) (+)(+) (+) (+)(-) negative 0und.

132. Slide 2- 132 Example Solving an Inequality Involving a Radical

133. Slide 2- 133 Example Solving an Inequality Involving a Radical 2 (-)(+)(+)(+) undefined positivenegative 00

134. Slide 2- 134 Chapter Test

135. Slide 2- 135 Chapter Test

136. Slide 2- 136 Chapter Test

137. Slide 2- 137 Chapter Test Solutions

138. Slide 2- 138 Chapter Test Solutions

139. Slide 2- 139 Chapter Test Solutions