1. Linear and Quadratic Functions and Modeling
2.1
Linear and Quadratic Functions and Modeling

2. Quick Review

3. Quick Review Solutions

4. 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.

5. Polynomial Function

6. Polynomial Functions of No and Low Degree
Name Form Degree
Zero Function f(x)=0 Undefined
Constant Function f(x)=a (a≠0) 0
Linear Function f(x)=ax+b (a≠0) 1
Quadratic Function f(x)=ax2+bx+c (a≠0) 2

7. Example Finding an Equation of a Linear Function

8. Example Finding an Equation of a Linear Function

9. Average Rate of Change

10. 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.

11. Characterizing the Nature of a Linear Function
Point of View Characterization
Verbal polynomial of degree 1
Algebraic f(x) = mx + b (m≠0)
Graphical slant line with slope m and y-intercept b
Analytical function 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

12. Properties of the Correlation Coefficient, r
When r > 0, there is a positive linear correlation.
When r < 0, there is a negative linear correlation.
When |r| ≈ 1, there is a strong linear correlation.
When |r| ≈ 0, there is weak or no linear correlation.

13. Linear Correlation

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

15. Example Transforming the Squaring Function

16. Example Transforming the Squaring Function

17. The Graph of f(x)=ax2

18. Vertex Form of a Quadratic Equation
Any quadratic function f(x) = ax2 + 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 – ah2. If a>0, the parabola opens upward, and if a<0, it opens downward.

19. Example Finding the Vertex and Axis of a Quadratic Function

20. Example Finding the Vertex and Axis of a Quadratic Function

21. Characterizing the Nature of a Quadratic Function
Point of View Characterization

22. Vertical Free-Fall Motion

23. Power Functions and Modeling
2.2
Power Functions and Modeling

24. Quick Review

25. Quick Review Solutions

26. 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.

27. Power Function
Any function that can be written in the form
f(x) = k·xa, 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 ath power of x, or f(x) is proportional to the
ath power of x.

28. Example Analyzing Power Functions

29. Example Analyzing Power Functions

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

31. Example Graphing Monomial Functions

32. Example Graphing Monomial Functions

33. Graphs of Power Functions
For any power function f(x) = k·xa, 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.

34. Graphs of Power Functions

35. Polynomial Functions of Higher Degree with Modeling
2.3
Polynomial Functions of Higher Degree with Modeling

36. Quick Review

37. Quick Review Solutions

38. 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.

39. The Vocabulary of Polynomials

40. Example Graphing Transformations of Monomial Functions

41. Example Graphing Transformations of Monomial Functions

42. Cubic Functions

43. Quartic Function

44. 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.

45. Leading Term Test for Polynomial End Behavior

46. Example Applying Polynomial Theory

47. Example Applying Polynomial Theory

48. Example Finding the Zeros of a Polynomial Function

49. Example Finding the Zeros of a Polynomial Function

50. Multiplicity of a Zero of a Polynomial Function

51. Example Sketching the Graph of a Factored Polynomial

52. Example Sketching the Graph of a Factored Polynomial

53. 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 y0 is between f(a) and f(b), then y0=f(c)
for some number c in [a,b].

54. Real Zeros of Polynomial Functions
2.4
Real Zeros of Polynomial Functions

55. Quick Review

56. Quick Review Solutions

57. 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.

58. Division Algorithm for Polynomials

59. Example Using Polynomial Long Division

60. Example Using Polynomial Long Division

61. Remainder Theorem

62. Example Using the Remainder Theorem

63. Example Using the Remainder Theorem

64. Factor Theorem

65. Example Using Synthetic Division

66. Example Using Synthetic Division

67. Rational Zeros Theorem

68. Upper and Lower Bound Tests for Real Zeros

69. Example Finding the Real Zeros of a Polynomial Function

70. Example Finding the Real Zeros of a Polynomial Function

71. Example Finding the Real Zeros of a Polynomial Function

72. Example Finding the Real Zeros of a Polynomial Function

73. Complex Zeros and the Fundamental Theorem of Algebra
2.5
Complex Zeros and the Fundamental Theorem of Algebra

74. Quick Review

75. Quick Review Solutions

76. 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.

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

78. Linear Factorization Theorem

79. 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).

80. Example Exploring Fundamental Polynomial Connections

81. Example Exploring Fundamental Polynomial Connections

82. Complex Conjugate Zeros

83. Example Finding a Polynomial from Given Zeros

84. Example Finding a Polynomial from Given Zeros

85. 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.

86. Example Factoring a Polynomial

87. Example Factoring a Polynomial

88. Graphs of Rational Functions
2.6
Graphs of Rational Functions

89. Quick Review

90. Quick Review Solutions

91. 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.

92. Rational Functions

93. Example Finding the Domain of a Rational Function

94. Example Finding the Domain of a Rational Function

95. Graph a Rational Function

96. Graph a Rational Function

97. Example Finding Asymptotes of Rational Functions

98. Example Finding Asymptotes of Rational Functions

99. Example Graphing a Rational Function

100. Example Graphing a Rational Function

101. Solving Equations in One Variable
2.7
Solving Equations in One Variable

102. Quick Review

103. Quick Review Solutions

104. 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.

105. 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.

106. Example Solving by Clearing Fractions

107. Example Solving by Clearing Fractions

108. Example Eliminating Extraneous Solutions

109. Example Eliminating Extraneous Solutions

110. Example Finding a Minimum Perimeter

111. Example Finding a Minimum Perimeter

112. Solving Inequalities in One Variable
2.8
Solving Inequalities in One Variable

113. Quick Review

114. Quick Review Solutions

115. 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.

116. Polynomial Inequalities

117. Example Finding where a Polynomial is Zero, Positive, or Negative

118. Example Finding where a Polynomial is Zero, Positive, or Negative-3 4
(-)(-)2
(+)(-)2
(+)(+)2
negative
positive

119. Example Solving a Polynomial Inequality Graphically

120. Example Solving a Polynomial Inequality Graphically

121. Example Creating a Sign Chart for a Rational Function

122. Example Creating a Sign Chart for a Rational Function-3 1
(-)
(-)(-)
negative
positive -1
(+)(-)
(+)
(+)(+)
und.

123. Example Solving an Inequality Involving a Radical

124. Example Solving an Inequality Involving a Radical-1 2
(-)(+)
(+)(+)
undefined
positive
negative

125. Chapter Test

126. Chapter Test

127. Chapter Test

128. Chapter Test Solutions

129. Chapter Test Solutions

130. Chapter Test Solutions