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.

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 Remainder Check: Quotient * Divisor + Remainder = Dividend

68. Slide 2- 68 Division Algorithm for Polynomials

69. Slide 2- 69 Dividing Polynomials Long division of polynomials is similar to long division of whole numbers. dividend = (quotient divisor) + remainder The result is written in the form: quotient + When you divide two polynomials you can check the answer using the following:

70. Slide 2- 70 + 2 Example: Divide x 2 + 3x – 2 by x + 1 and check the answer. x x 2 + x 2x2x– 2 2x + 2 – 4– 4 remainder Check: 1. 2. 3. 4. 5. 6. correct (x + 2) quotient (x + 1) divisor + (– 4) remainder = x 2 + 3x – 2 dividend Answer: x + 2 + – 4– 4

71. Slide 2- 71 Example: Divide 4x + 2x 3 – 1 by 2x – 2 and check the answer. Write the terms of the dividend in descending order. 1. x2x2 2. 2x 3 – 2x 2 3. 2x22x2 + 4x 4. + x 5. 2x 2 – 2x 6. 6x6x – 1 7. + 3 8. 6x – 6 9. 5 Check: (x 2 + x + 3)(2x – 2) + 5 = 4x + 2x 3 – 1 Answer: x 2 + x + 3 5 Since there is no x 2 term in the dividend, add 0x 2 as a placeholder.

72. Slide 2- 72 x x 2 – 2x – 3x + 6 – 3 – 3x + 6 0 Answer: x – 3 with no remainder. Check: (x – 2)(x – 3) = x 2 – 5x + 6 Example: Divide x 2 – 5x + 6 by x – 2.

73. Slide 2- 73 Example: Divide x 3 + 3x 2 – 2x + 2 by x + 3 and check the answer. x2x2 x 3 + 3x 2 0x20x2 – 2x – 2 – 2x – 6 8 Check: (x + 3)(x 2 – 2) + 8 = x 3 + 3x 2 – 2x + 2 Answer: x 2 – 2 + 8 + 2 Note: the first subtraction eliminated two terms from the dividend. Therefore, the quotient skips a term. + 0x

74. Slide 2- 74 16 Synthetic division is a shorter method of dividing polynomials. This method can be used only when the divisor is of the form x – a. It uses the coefficients of each term in the dividend. Example: Divide 3x 2 + 2x – 1 by x – 2 using synthetic division. 3 2 – 1 2 Since the divisor is x – 2, a = 2. 3 1. Bring down 3 2. (2 3) = 6 6 815 3. (2 + 6) = 8 4. (2 8) = 16 5. (–1 + 16) = 15 coefficients of quotient remainder value of a coefficients of the dividend 3x + 8Answer: 15

75. Slide 2- 75 Example: Divide x 3 – 3x + 4 by x + 3 using synthetic division. Notice that the degree of the first term of the quotient is one less than the degree of the first term of the dividend. remainder a coefficients of quotient – 3– 3 Since, x – a = x + 3, a = – 3. 1 0 – 3 4 1 – 3– 3 – 3– 3 9– 18 6– 14 coefficients of dividend = x 2 – 3x + 6 – 14 Insert zero coefficient as placeholder for the missing x 2 term.

76. Slide 2- 76 Remainder Theorem: The remainder of the division of a polynomial f (x) by x – a is f (a). Example: Using the remainder theorem, evaluate f(x) = x 4 – 4x – 1 when x = 3. 9 1 0 0 – 4 – 1 3 1 3 39 6927 2368 The remainder is 68 at x = 3, so f (3) = 68. You can check this using substitution:f(3) = (3) 4 – 4(3) – 1 = 68. value of x

77. Slide 2- 77 Example: Using synthetic division and the remainder theorem, evaluate f (x) = x 2 – x at x = – 2. 6 1 – 1 0 – 2– 2 1 – 2– 2 – 3– 36 Then f (– 2) = 6 and (– 2, 6) is a point on the graph of f(x) = x 2 – x. f(x) = x 2 – x x y 2 4 (– 2, 6) remainder

78. Slide 2- 78 Example Using Polynomial Long Division

79. Slide 2- 79 Example Using Polynomial Long Division

80. Slide 2- 80 Remainder Theorem

81. Slide 2- 81 Example Using the Remainder Theorem

82. Slide 2- 82 Example Using the Remainder Theorem

83. Slide 2- 83 Factor Theorem

84. Slide 2- 84 Example Using Synthetic Division

85. Slide 2- 85 Rational Zeros Theorem

86. Slide 2- 86 Upper and Lower Bound Tests for Real Zeros

87. Slide 2- 87 Show 2x - 3 is a factor of 6x 2 + x – 15 (x = 3/2) 3/2 6 1 -15 6 9 10 15 0 Note: Since the remainder is 0, 2x - 3 is a factor of 6x 2 + x – 15

88. Slide 2- 88 Factor x 4 – 3x 3 – 5x 2 + 3x + 4 are possible factors P(1) = 0 1 1 -3 -5 3 4 1 1 -2 -7 -4 0 x 3 – 2x 2 – 7x - 4 so x 4 – 3x 3 – 5x 2 + 3x + 4 = (x 3 – 2x 2 – 7x – 4)(x – 1)

89. Slide 2- 89 Factor x 3 – 2x 2 – 7x - 4 are possible factors P(-1) = 0 -1 1 -2 -7 -4 1 -3 3 -4 4 0 x 2 – 3x – 4 so x 4 – 3x 3 – 5x 2 + 3x + 4 = (x 2 – 3x – 4)(x – 1)(x + 1)

90. Slide 2- 90 Factor x 3 – 2x 2 – 7x - 4 are possible factors P(-1) = 0 -1 1 -2 -7 -4 1 -3 3 -4 4 0 x 2 – 3x – 4 so x 4 – 3x 3 – 5x 2 + 3x + 4 = (x 2 – 3x – 4)(x – 1)(x + 1)

91. Slide 2- 91 x 4 – 3x 3 – 5x 2 + 3x + 4 = (x 2 – 3x – 4)(x – 1)(x + 1) or (x – 4)(x + 1)(x – 1)(x + 1)

92. Slide 2- 92 Factor x 4 – 8x 3 +17x 2 + 2x - 24 are possible factors P(4) = 0 4 1 -8 17 2 -24 1 4 -4 -16 1 4 6 24 0 x 3 – 4x 2 + x + 6 so x 4 – 8x 3 + 17x 2 + 2x - 24 = (x 3 – 4x 2 + x + 6)(x – 4)

93. Slide 2- 93 Factor x 3 – 4x 2 + x + 6 P(2) = 0 2 1 -4 1 6 1 2 -2 -4 -3 -6 0 x 2 – 2x – 3 so x 4 – 8x 3 +17x 2 + 2x - 24 = (x 2 – 2x – 3)(x – 4)(x - 2) are possible factors

94. Slide 2- 94 x 4 – 8x 3 +17x 2 + 2x - 24 = (x 2 – 2x – 3)(x – 4)(x - 2) or (x – 3)(x + 1)(x – 4)(x - 2)

95. Slide 2- 95 Show x 3 – 3x 2 + 5 = 0 has no rational roots  1,  5 only possible roots P(1) = 1 3 – 3(1) 2 + 5 = 3 P(-1) = (-1) 3 – 3(-1) 2 + 5 = 1 P(5) = (5) 3 – 3(5) 2 + 5 = 55 P(-5) = (-5) 3 – 3(-5) 2 + 5 = -195 Since none of the possible roots give zero in the remainder theorem, there are no rational roots.

96. Slide 2- 96 Example Finding the Real Zeros of a Polynomial Function

97. Slide 2- 97 Example Finding the Real Zeros of a Polynomial Function

98. Slide 2- 98 Example Finding the Real Zeros of a Polynomial Function

99. Slide 2- 99 Example Finding the Real Zeros of a Polynomial Function

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

101. Slide 2- 101 Quick Review

102. Slide 2- 102 Quick Review Solutions

103. Slide 2- 103 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.

104. Slide 2- 104 Fundamental Theorem of Algebra A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

105. Slide 2- 105 Linear Factorization Theorem

106. Slide 2- 106 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).

107. Slide 2- 107 Example Exploring Fundamental Polynomial Connections

108. Slide 2- 108 Example Exploring Fundamental Polynomial Connections

109. Slide 2- 109 Complex Conjugate Zeros

110. Slide 2- 110 Example Finding a Polynomial from Given Zeros

111. Slide 2- 111 Example Finding a Polynomial from Given Zeros

112. Slide 2- 112 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.

113. Slide 2- 113 Find an equation of a polynomial with roots of 2i and 3. (x – 2i)(x + 2i)(x – 3) = 0 (x 2 – 2ix + 2ix – 4i 2 )(x – 3) = 0 (x 2 + 4)(x – 3) = 0 x 3 – 3x 2 + 4x – 12 = 0

114. Slide 2- 114 Find an equation of a polynomial with roots of 1- i and -2. (x – (1 - i))(x – (1 + i))(x + 2) = 0 (x 2 – x - ix – x + 1 + i +ix -i - i 2 )(x + 2) = 0 (x 2 – 2x + 2)(x + 2) = 0 x 3 + 2x 2 –2x 2 - 4x + 2x + 4= 0 (x – 1 + i)(x – 1 - i)(x + 2) = 0 x 3 - 2x + 4 = 0

115. Slide 2- 115 2.5 Complex Zeros and the Fundamental Theorem of Algebra Page 230

116. Slide 2- 116 2.5 Complex Zeros and the Fundamental Theorem of Algebra (cont’d) Page 230

117. Slide 2- 117 Example Factoring a Polynomial

118. Slide 2- 118 Example Factoring a Polynomial

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

120. Slide 2- 120 Quick Review

121. Slide 2- 121 Quick Review Solutions

122. Slide 2- 122 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.

123. Slide 2- 123 Rational Functions

124. Slide 2- 124 Example Finding the Domain of a Rational Function

125. Slide 2- 125 Example Finding the Domain of a Rational Function

126. Slide 2- 126 3 Rules for Asymptotes for Rational Functions

127. Slide 2- 127 Graph a Rational Function

128. Slide 2- 128 Graph a Rational Function

129. Slide 2- 129 Example Finding Asymptotes of Rational Functions

130. Slide 2- 130 Example Finding Asymptotes of Rational Functions

131. Slide 2- 131 Example Graphing a Rational Function

132. Slide 2- 132 Example Graphing a Rational Function -3 1 2 -----  ++++++ 0 ----  ++++++

133. Slide 2- 133 Sketch the graph of f(0) = -3 Solve x – 3 = 0; x = 3 Solve x + 1 = 0; x = -1 and sketch the asymptote. Horizontal asymptote, y = 1 -3 1 +++  --------- 0 ++++++++ Example Graphing a Rational Function

134. Slide 2- 134 Sketch the graph of f(0) = -2 Solve x + 2 = 0; x = -2 Solve x 2 - 1 = 0; x = -1 x = 1 and sketch the asymptotes. Horizontal asymptote, y = 0 -2 -1 1 ----  +++ 0 ----  ++++++ Example Graphing a Rational Function

135. Slide 2- 135 Graphs of Rational Functions Page 239

136. Slide 2- 136 Graphs of Rational Functions Page 239

137. Slide 2- 137 Graphs of Rational Functions Page 239

138. Slide 2- 138 Graphs of Rational Functions Page 239

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

140. Slide 2- 140 Quick Review

141. Slide 2- 141 Quick Review Solutions

142. Slide 2- 142 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.

143. Slide 2- 143 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.

144. Slide 2- 144 Example Solving by Clearing Fractions

145. Slide 2- 145 Example Solving by Clearing Fractions The LCD is x Multiply both sides by x Subtract 3x from both sides (x – 2)(x – 1) = 0 x = 2 or x = 1 Check Both solutions are correct

146. Slide 2- 146 Example Eliminating Extraneous Solutions The LCD is (x – 1)(x - 3) (x – 1)(1) + 2x(x - 3) = 2 2x 2 – 5x – 3 = 0 (2x + 1)(x - 3) = 0 x = -½ or x = 3 Check: x = 3 is not defined x = -½ is the only solution x – 1 + 2x 2 - 6x = 2

147. Slide 2- 147 Example Eliminating Extraneous Solutions The LCD is (x)(x + 2) (x + 2)(x - 3) + 3x + 6 = 0 x 2 + 2x = 0 x(x + 2) = 0 x = 0 or x = -2 Check: x = 0 is not defined x = -2 is not defined No Solution x 2 – x – 6 + 3x + 6 = 0 Solve the equation

148. Slide 2- 148 Example Finding a Minimum Perimeter

149. Slide 2- 149 Example Acid Problem a.Pure acid is added to 78 oz of a 63% acid solution. Let x be the amount (in ounces) of pure acid added. Find an algebraic representation for C(x), the concentration of acid as a function of x. Determine how much pure acid should be added so that the mixture is at least 83% acid. 1.00x + (.63)(78) = C(x)(x + 78) 1.00x + 49.14 >.83x + 64.74.17x > 15.6x > 91.76

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

151. Slide 2- 151 Quick Review Solutions

152. Slide 2- 152 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.

153. Slide 2- 153 Polynomial Inequalities

154. Slide 2- 154 Example Finding where a Polynomial is Zero, Positive, or Negative

155. Slide 2- 155 Example Finding where a Polynomial is Zero, Positive, or Negative -34 (-)(-) 2 (+)(-) 2 (+)(+) 2 negative positive

156. Slide 2- 156 Example Solving a Polynomial Inequality Graphically

157. Slide 2- 157 Example Solving a Polynomial Inequality Graphically

158. Slide 2- 158 Example Creating a Sign Chart for a Rational Function

159. Slide 2- 159 Example Creating a Sign Chart for a Rational Function -31 (-) (-)(-) negative positive (-) (+)(-) (+) (+)(+) (+) (+)(-) negative 0und. 0

160. Slide 2- 160 Example Solving an Inequality Involving a Radical

161. Slide 2- 161 Example Solving an Inequality Involving a Radical 2 (-)(+)(+)(+) undefined positivenegative 00

162. Slide 2- 162 Chapter Test

163. Slide 2- 163 Chapter Test

164. Slide 2- 164 Chapter Test

165. Slide 2- 165 Chapter Test Solutions

166. Slide 2- 166 Chapter Test Solutions

167. Slide 2- 167 Chapter Test Solutions