Polynomial and Rational Functions

1
Polynomial and Rational Functions

• Chapter 3

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2
Quadratic Functions and Models

• Section 3.1

3
Quadratic Functions

• Quadratic function Function of the form
• f(x) ax2 bx c
• (a, b and c real numbers, a ? 0)

4
Quadratic Functions

• Example. Plot the graphs of f(x) x2, g(x) 3x2
  and

5
Quadratic Functions

• Example. Plot the graphs of f(x) x2, g(x)
  3x2 and

6
Parabolas

• Parabola The graph of a quadratic function
• If a gt 0, the parabola opens up
• If a lt 0, the parabola opens down
• Vertex highest / lowest point of a parabola

7
Parabolas

• Axis of symmetry Vertical line passing through
  the vertex

8
Parabolas

• Example. For the function
• f(x) 3x2 12x 11
• (a) Problem Graph the function
• Answer

9
Parabolas

• Example. (cont.)
• (b) Problem Find the vertex and axis of
  symmetry.
• Answer

10
Parabolas

• Locations of vertex and axis of symmetry
• Set
• Set
• Vertex is at
• Axis of symmetry runs through vertex

11
Parabolas

• Example. For the parabola defined by
• f(x) 2x2 3x 2
• (a) Problem Without graphing, locate the vertex.
• Answer
• (b) Problem Does the parabola open up or down?
• Answer

12
x-intercepts of a Parabola

• For a quadratic function f(x) ax2 bx c
• Discriminant is b2 4ac.
• Number of x-intercepts depends on the
  discriminant.
• Positive discriminant Two x-intercepts
• Negative discriminant Zero x-intercepts
• Zero discriminant One x-intercept (Vertex lies
  on x-axis)

13
x-intercepts of a Parabola
14
Graphing Quadratic Functions

• Example. For the function
• f(x) 2x2 8x 4
• (a) Problem Find the vertex
• Answer
• (b) Problem Find the intercepts.
• Answer

15
Graphing Quadratic Functions

• Example. (cont.)
• (c) Problem Graph the function
• Answer

16
Graphing Quadratic Functions

• Example. (cont.)
• (d) Problem Determine the domain and range of f.
• Answer
• (e) Problem Determine where f is increasing and
  decreasing.
• Answer

17
Graphing Quadratic Functions

• Example.
• Problem Determine the quadratic function whose
  vertex is (2, 3) and whose y-intercept is 11.
• Answer

18
Graphing Quadratic Functions

• Method 1 for Graphing
• Complete the square in x to write the quadratic
  function in the form y a(x h)2 k
• Graph the function using transformations

19
Graphing Quadratic Functions

• Method 2 for Graphing
• Determine the vertex
• Determine the axis of symmetry
• Determine the y-intercept f(0)
• Find the discriminant b2 4ac.
• If b2 4ac gt 0, two x-intercepts
• If b2 4ac 0, one x-intercept (at the
  vertex)
• If b2 4ac lt 0, no x-intercepts.

20
Graphing Quadratic Functions

• Method 2 for Graphing
• Find an additional point
• Use the y-intercept and axis of symmetry.
• Plot the points and draw the graph

21
Graphing Quadratic Functions

• Example. For the quadratic function
• f(x) 3x2 12x 7
• (a) Problem Determine whether f has a maximum or
  minimum value, then find it.
• Answer

22
Graphing Quadratic Functions

• Example. (cont.)
• (b) Problem Graph f
• Answer

23
Quadratic Relations
24
Quadratic Relations

• Example. An engineer collects the following data
  showing the speed s of a Ford Taurus and its
  average miles per gallon, M.

25
Quadratic Relations
Speed, s Miles per Gallon, M
30 18
35 20
40 23
40 25
45 25
50 28
55 30
60 29
65 26
65 25
70 25
26
Quadratic Relations

• Example. (cont.)
• (a) Problem Draw a scatter diagram of the data
• Answer

27
Quadratic Relations

• Example. (cont.)
• (b) Problem Find the quadratic function of best
  fit to these data.
• Answer

28
Quadratic Relations

• Example. (cont.)
• (c) Problem Use the function to determine the
  speed that maximizes miles per gallon.
• Answer

29
Key Points

• Quadratic Functions
• Parabolas
• x-intercepts of a Parabola
• Graphing Quadratic Functions
• Quadratic Relations

30
Polynomial Functions and Models

• Section 3.2

31
Polynomial Functions

• Polynomial function Function of the form
• f(x) anxn an 1xn 1 ??? a1x a0
• an, an 1, , a1, a0 real numbers
• n is a nonnegative integer (an ? 0)
• Domain is the set of all real numbers
• Terminology
• Leading coefficient an
• Degree n (largest power)
• Constant term a0

32
Polynomial Functions

• Degrees
• Zero function undefined degree
• Constant functions degree 0.
• (Non-constant) linear functions degree 1.
• Quadratic functions degree 2.

33
Polynomial Functions

• Example. Determine which of the following are
  polynomial functions? For those that are, find
  the degree.
• (a) Problem f(x) 3x 6x2
• Answer
• (b) Problem g(x) 13x3 5 9x4
• Answer
• (c) Problem h(x) 14
• Answer
• (d) Problem
• Answer

34
Polynomial Functions

• Graph of a polynomial function will be smooth and
  continuous.
• Smooth no sharp corners or cusps.
• Continuous no gaps or holes.

35
Power Functions

• Power function of degree n
• Function of the form
• f(x) axn
• a ? 0 a real number
• n gt 0 is an integer.

36
Power Functions

• The graph depends on whether n is even or odd.

37
Power Functions

• Properties of f(x) axn
• Symmetry
• If n is even, f is even.
• If n is odd, f is odd.
• Domain All real numbers.
• Range
• If n is even, All nonnegative real numbers
• If n is odd, All real numbers.

38
Power Functions

• Properties of f(x) axn
• Points on graph
• If n is even (0, 0), (1, 1) and (1, 1)
• If n is odd (0, 0), (1, 1) and (1, 1)
• Shape As n increases
• Graph becomes more vertical if x gt 1
• More horizontal near origin

39
Graphing Using Transformations

• Example.
• Problem Graph f(x) (x 1)4
• Answer

40
Graphing Using Transformations

• Example.
• Problem Graph f(x) x5 2
• Answer

41
Zeros of a Polynomial

• Zero or root of a polynomial f
• r a real number for which f(r) 0
• r is an x-intercept of the graph of f.
• (x r) is a factor of f.

42
Zeros of a Polynomial
43
Zeros of a Polynomial

• Example.
• Problem Find a polynomial of degree 3 whose
  zeros are 4, 2 and 3.
• Answer

44
Zeros of a Polynomial

• Repeated or multiple zero or root of f
• Same factor (x r) appears more than once
• Zero of multiplicity m
• (x r)m is a factor of f and (x r)m1 isnt.

45
Zeros of a Polynomial

• Example.
• Problem For the polynomial, list all zeros and
  their multiplicities.
• f(x) 2(x 2)(x 1)3(x 3)4
• Answer

46
Zeros of a Polynomial

• Example. For the polynomial
• f(x) x3(x 3)2(x 2)
• (a) Problem Graph the polynomial
• Answer

47
Zeros of a Polynomial

• Example. (cont.)
• (b) Problem Find the zeros and their
  multiplicities
• Answer

48
Multiplicity

• Role of multiplicity
• r a zero of even multiplicity
• f(x) does not change sign at r
• Graph touches the x-axis at r, but does not cross

49
Multiplicity

• Role of multiplicity
• r a zero of odd multiplicity
• f(x) changes sign at r
• Graph crosses x-axis at r

50
Turning Points

• Turning points
• Points where graph changes from increasing to
  decreasing function or vice versa
• Turning points correspond to local extrema.
• Theorem. If f is a polynomial function of degree
  n, then f has at most n 1 turning points.

51
End Behavior

• Theorem. End Behavior
• For large values of x, either positive or
  negative, that is, for large x, the graph of
  the polynomial
• f(x) anxn an1xn1 ? a1x a0
• resembles the graph of the power function
• y anxn

52
End Behavior

• End behavior of
• f(x) anxn an1xn1 ? a1x a0

53
Analyzing Polynomial Graphs

• Example. For the polynomial
• f(x) 12x3 2x4 2x5
• (a) Problem Find the degree.
• Answer
• (b) Problem Determine the end behavior. (Find
  the power function that the graph of f resembles
  for large values of x.)
• Answer

54
Analyzing Polynomial Graphs

• Example. (cont.)
• (c) Problem Find the x-intercept(s), if any
• Answer
• (d) Problem Find the y-intercept.
• Answer
• (e) Problem Does the graph cross or touch the
  x-axis at each x-intercept
• Answer

55
Analyzing Polynomial Graphs

• Example. (cont.)
• (f) Problem Graph f using a graphing utility
• Answer

56
Analyzing Polynomial Graphs

• Example. (cont.)
• (g) Problem Determine the number of turning
  points on the graph of f. Approximate the turning
  points to 2 decimal places.
• Answer
• (h) Problem Find the domain
• Answer

57
Analyzing Polynomial Graphs

• Example. (cont.)
• (i) Problem Find the range
• Answer
• (j) Problem Find where f is increasing
• Answer
• (k) Problem Find where f is decreasing
• Answer

58
Cubic Relations
59
Cubic Relations

• Example. The following data represent the average
  number of miles driven (in thousands) annually by
  vans, pickups, and sports utility vehicles for
  the years 1993-2001, where x 1 represents 1993,
  x 2 represents 1994, and so on.

60
Cubic Relations
Year, x Average Miles Driven, M
1993, 1 12.4
1994, 2 12.2
1995, 3 12.0
1996, 4 11.8
1997, 5 12.1
1998, 6 12.2
1999, 7 12.0
2000, 8 11.7
2001, 9 11.1
61
Cubic Relations

• Example. (cont.)
• (a) Problem Draw a scatter diagram of the data
  using x as the independent variable and M as the
  dependent variable.
• Answer

62
Cubic Relations

• Example. (cont.)
• (b) Problem Find the cubic function of best fit
  and graph it
• Answer

63
Key Points

• Polynomial Functions
• Power Functions
• Graphing Using Transformations
• Zeros of a Polynomial
• Multiplicity
• Turning Points
• End Behavior
• Analyzing Polynomial Graphs
• Cubic Relations

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The Real Zeros of a Polynomial Function

• Section 3.6

65
Division Algorithm

• Theorem. Division AlgorithmIf f(x) and g(x)
  denote polynomial functions and if g(x) is a
  polynomial whose degree is greater than zero,
  then there are unique polynomial functions q(x)
  and r(x) such that
• where r(x) is either the zero polynomial or a
  polynomial of degree less than that of g(x).

66
Division Algorithm

• Division algorithm
• f(x) is the dividend
• q(x) is the quotient
• g(x) is the divisor
• r(x) is the remainder

67
Remainder Theorem

• First-degree divisor
• Has form g(x) x c
• Remainder r(x)
• Either the zero polynomial or a polynomial of
  degree 0,
• Either way a number R.
• Becomes f(x) (x c)q(x) R
• Substitute x c
• Becomes f(c) R

68
Remainder Theorem

• Theorem. Remainder Theorem Let f be a
  polynomial function. If f(x) is divided by x c,
  the remainder is f(c).

69
Remainder Theorem

• Example. Find the remainder if
• f(x) x3 3x2 2x 6
• is divided by
• (a) Problem x 2
• Answer
• (b) Problem x 1
• Answer

70
Factor Theorem

• Theorem. Factor Theorem Let f be a polynomial
  function. Then x c is a factor of f(x) if and
  only if f(c) 0.
• If f(c) 0, then x c is a factor of f(x).
• If x c is a factor of f(x), then f(c) 0.

71
Factor Theorem

• Example. Determine whether the function
• f(x) 2x3 x2 4x 3
• has the given factor
• (a) Problem x 1
• Answer
• (b) Problem x 1
• Answer

72
Number of Real Zeros

• Theorem. Number of Real ZerosA polynomial
  function of degree n, n 1, has at most n real
  zeros.

73
Rational Zeros Theorem

• Theorem. Rational Zeros TheoremLet f be a
  polynomial function of degree 1 or higher of the
  form
• f(x) anxn an1xn1 ? a1x a0
• an ? 0, a0 ? 0, where each coefficient is an
  integer. If p/q, in lowest terms, is a rational
  zero of f, then p must be a factor of a0 and q
  must be a factor of an.

74
Rational Zeros Theorem

• Example.
• Problem List the potential rational zeros of
• f(x) 3x3 8x2 7x 12
• Answer

75
Finding Zeros of a Polynomial

• Determine the maximum number of zeros.
• Degree of the polynomial
• If the polynomial has integer coefficients
• Use the Rational Zeros Theorem to find potential
  rational zeros
• Using a graphing utility, graph the function.

76
Finding Zeros of a Polynomial

• Test values
• Test a potential rational zero
• Each time a zero is found, repeat on the
  depressed equation.

77
Finding Zeros of a Polynomial

• Example.
• Problem Find the rational zeros of the
  polynomial in the last example.
• f(x) 3x3 8x2 7x 12
• Answer

78
Finding Zeros of a Polynomial

• Example.
• Problem Find the real zeros of
• f(x) 2x4 13x3 29x2 27x 9
• and write f in factored form
• Answer

79
Factoring Polynomials

• Irreducible quadratic Cannot be factored over
  the real numbers
• Theorem. Every polynomial function (with real
  coefficients) can be uniquely factored into a
  product of linear factors and irreducible
  quadratic factors
• Corollary. A polynomial function (with real
  coefficients) of odd degree has at least one real
  zero

80
Factoring Polynomials

• Example.
• Problem Factor
• f(x)2x5 9x4 20x3 40x2 48x 16
• Answer

81
Bounds on Zeros

• Bound on the zeros of a polynomial
• Positive number M
• Every zero lies between M and M.

82
Bounds on Zeros

• Theorem. Bounds on ZerosLet f denote a
  polynomial whose leading coefficient is 1.
• f(x) xn an1xn1 ? a1x a0
• A bound M on the zeros of f is the smaller of
  the two numbers
• Max1, ja0j ja1j ? jan-1j,
• 1 Maxja0j ,ja1j , , jan-1j

83
Bounds on Zeros

• Example. Find a bound to the zeros of each
  polynomial.
• (a) Problem
• f(x) x5 6x3 7x2 8x 10
• Answer
• (b) Problem
• g(x) 3x5 4x4 2x3 x2 5
• Answer

84
Intermediate Value Theorem

• Theorem. Intermediate Value TheoremLet f
  denote a continuous function. If a lt b and if
  f(a) and f(b) are of opposite sign, then f has at
  least one zero between a and b.

85
Intermediate Value Theorem

• Example.
• Problem Show that
• f(x) x5 x4 7x3 7x2 18x 18
• has a zero between 1.4 and 1.5. Approximate it
  to two decimal places.
• Answer

86
Key Points

• Division Algorithm
• Remainder Theorem
• Factor Theorem
• Number of Real Zeros
• Rational Zeros Theorem
• Finding Zeros of a Polynomial
• Factoring Polynomials
• Bounds on Zeros
• Intermediate Value Theorem

87
Complex Zeros Fundamental Theorem of Algebra

• Section 3.7

88
Complex Polynomial Functions

• Complex polynomial function Function of the form
• f(x) anxn an 1xn 1 ??? a1x a0
• an, an 1, , a1, a0 are all complex numbers,
• an ? 0,
• n is a nonnegative integer
• x is a complex variable.
• Leading coefficient of f an
• Complex zero A complex number r with f(r) 0.

89
Complex Arithmetic

• See Appendix A.6.
• Imaginary unit Number i with i2 1.
• Complex number Number of the form z a bi
• a and b real numbers.
• a is the real part of z
• b is the imaginary part of z
• Can add, subtract, multiply
• Can also divide (we wont)

90
Complex Arithmetic

• Conjugate of the complex number a bi
• Number a bi
• Written
• Properties

91
Complex Arithmetic

• Example. Suppose z 5 2i and w 2 3i.
• (a) Problem Find z w
• Answer
• (b) Problem Find z w
• Answer
• (c) Problem Find zw
• Answer
• (d) Problem Find
• Answer

92
Fundamental Theorem of Algebra

• Theorem. Fundamental Theorem of AlgebraEvery
  complex polynomial function f(x) of degree n 1
  has at least one complex zero.

93
Fundamental Theorem of Algebra

• Theorem. Every complex polynomial function f(x)
  of degree n 1 can be factored into n linear
  factors (not necessarily distinct) of the form
• f(x) an(x r1)(x r2) ? (x rn)
• where an, r1, r2, , rn are complex numbers.
  That is, every complex polynomial function f(x)
  of degree n 1 has exactly n (not necessarily
  distinct) zeros.

94
Conjugate Pairs Theorem

• Theorem. Conjugate Pairs TheoremLet f(x) be a
  polynomial whose coefficients are real numbers.
  If a bi is a zero of f, then the complex
  conjugate a bi is also a zero of f.

95
Conjugate Pairs Theorem

• Example. A polynomial of degree 5 whose
  coefficients are real numbers has the zeros 2,
  3i and 2 4i.
• Problem Find the remaining two zeros.
• Answer

96
Conjugate Pairs Theorem

• Example.
• Problem Find a polynomial f of degree 4 whose
  coefficients are real numbers and that has the
  zeros 2, 1 and 4 i.
• Answer

97
Conjugate Pairs Theorem

• Example.
• Problem Find the complex zeros of the polynomial
  function
• f(x) x4 2x3 x2 8x 20
• Answer

98
Key Points

• Complex Polynomial Functions
• Complex Arithmetic
• Fundamental Theorem of Algebra
• Conjugate Pairs Theorem

99
Properties of Rational Functions

• Section 3.3

100
Rational Functions

• Rational function Function of the form
• p and q are polynomials,
• q is not the zero polynomial.
• Domain Set of all real numbers except where q(x)
  0

101
Rational Functions

• is in lowest terms
• The polynomials p and q have no common factors
• x-intercepts of R
• Zeros of the numerator p when R is in lowest
  terms

102
Rational Functions

• Example. For the rational function
• (a) Problem Find the domain
• Answer
• (b) Problem Find the x-intercepts
• Answer
• (c) Problem Find the y-intercepts
• Answer

103
Graphing Rational Functions

• Graph of

104
Graphing Rational Functions

• As x approaches 0, is unbounded in the positive
  direction.
• Write f(x) ! 1
• Read f(x) approaches infinity
• Also
• May write f(x) ! 1 as x ! 0
• May read f(x) approaches infinity as x
  approaches 0

105
Graphing Rational Functions

• Example. For
• Problem Use transformations to graph f.
• Answer

106
Asymptotes

• Horizontal asymptotes
• Let R denote a function.
• Let x ! 1 or as x ! 1,
• If the values of R(x) approach some fixed number
  L, then the line y L is a horizontal asymptote
  of the graph of R.

107
Asymptotes

• Vertical asymptotes
• Let x ! c
• If the values jR(x)j ! 1, then the line x c is
  a vertical asymptote of the graph of R.

108
Asymptotes

• Asymptotes
• Oblique asymptote Neither horizontal nor
  vertical
• Graphs and asymptotes
• Graph of R never intersects a vertical asymptote.
• Graph of R can intersect a horizontal or oblique
  asymptote (but doesnt have to)

109
Asymptotes

• A rational function can have
• Any number of vertical asymptotes.
• 1 horizontal and 0 oblique asymptote
• 0 horizontal and 1 oblique asymptotes
• 0 horizontal and 0 oblique asymptotes
• There are no other possibilities

110
Vertical Asymptotes

• Theorem. Locating Vertical Asymptotes
• A rational function in lowest terms, will have
  a vertical asymptote x r if r is a real zero of
  the denominator q.

111
Vertical Asymptotes

• Example. Find the vertical asymptotes, if any, of
  the graph of each rational function.
• (a) Problem
• Answer
• (b) Problem
• Answer

112
Vertical Asymptotes

• Example. (cont.)
• (c) Problem
• Answer
• (d) Problem
• Answer

113
Horizontal and Oblique Asymptotes

• Describe the end behavior of a rational function.
• Proper rational function
• Degree of the numerator is less than the degree
  of the denominator.
• Theorem. If a rational function R(x) is proper,
  then y 0 is a horizontal asymptote of its graph.

114
Horizontal and Oblique Asymptotes

• Improper rational function R(x) one that is not
  proper.
• May be written
• where is proper. (Long division!)

115
Horizontal and Oblique Asymptotes

• If f(x) b, (a constant)
• Line y b is a horizontal asymptote
• If f(x) ax b, a ? 0,
• Line y ax b is an oblique asymptote
• In all other cases, the graph of R approaches the
  graph of f, and there are no horizontal or
  oblique asymptotes.
• This is all higher-degree polynomials

116
Horizontal and Oblique Asymptotes

• Example. Find the hoizontal or oblique
  asymptotes, if any, of the graph of each rational
  function.
• (a) Problem
• Answer
• (b) Problem
• Answer

117
Horizontal and Oblique Asymptotes

• Example. (cont.)
• (c) Problem
• Answer
• (d) Problem
• Answer

118
Key Points

• Rational Functions
• Graphing Rational Functions
• Vertical Asymptotes
• Horizontal and Oblique Asymptotes

119
The Graph of a Rational Function Inverse and
Joint Variation

• Section 3.4

120
Analyzing Rational Functions

• Find the domain of the rational function.
• Write R in lowest terms.
• Locate the intercepts of the graph.
• x-intercepts Zeros of numerator of function in
  lowest terms.
• y-intercept R(0), if 0 is in the domain.
• Test for symmetry Even, odd or neither.

121
Analyzing Rational Functions

• Locate the vertical asymptotes
• Zeros of denominator of function in lowest terms.
• Locate horizontal or oblique asymptotes
• Graph R using a graphing utility.
• Use the results obtained to graph by hand

122
Analyzing Rational Functions

• Example.
• Problem Analyze the graph of the rational
  function
• Answer
• Domain
• R in lowest terms
• x-intercepts
• y-intercept
• Symmetry

123
Analyzing Rational Functions

• Example. (cont.)
• Answer (cont.)
• Vertical asymptotes
• Horizontal asymptote
• Oblique asymptote

124
Analyzing Rational Functions

• Example. (cont.)
• Answer (cont.)

125
Analyzing Rational Functions

• Example.
• Problem Analyze the graph of the rational
  function
• Answer
• Domain
• R in lowest terms
• x-intercepts
• y-intercept
• Symmetry

126
Analyzing Rational Functions

• Example. (cont.)
• Answer (cont.)
• Vertical asymptotes
• Horizontal asymptote
• Oblique asymptote

127
Analyzing Rational Functions

• Example. (cont.)
• Answer (cont.)

128
Variation

• Inverse variation
• Let x and y denote 2 quantities.
• y varies inversely with x
• If there is a nonzero constant such that
• Also say y is inversely proportional to x

129
Variation

• Joint or Combined Variation
• Variable quantity Q proportional to the product
  of two or more other variables
• Say Q varies jointly with these quantities.
• Combinations of direct and/or inverse variation
  are combined variation.

130
Variation

• Example. Boyles law states that for a fixed
  amount of gas kept at a fixed temperature, the
  pressure P and volume V are inversely
  proportional (while one increases, the other
  decreases).  

131
Variation

• Example. According to Newton, the gravitational
  force between two objects varies jointly with the
  masses m1 and m2 of each object and inversely
  with the square of the distance r between the
  objects, hence

132
Key Points

• Analyzing Rational Functions
• Variation

133
Polynomial and Rational Inequalities

• Section 3.5

134
Solving Inequalities Algebraically

• Rewrite the inequality
• Left side Polynomial or rational expression f.
  (Write rational expression as a single quotient)
• Right side Zero
• Should have one of following forms
• f(x) gt 0
• f(x) 0
• f(x) lt 0
• f(x) 0

135
Solving Inequalities Algebraically

• Determine where left side is 0 or undefined.
• Separate the real line into intervals based on
  answers to previous step.

136
Solving Inequalities Algebraically

• Test Points
• Select a number in each interval
• Evaluate f at that number.
• If the value of f is positive, then f(x) gt 0
  for all numbers x in the interval.
• If the value of f is negative, then f(x) lt 0
  for all numbers x in the interval.

137
Solving Inequalities Algebraically

• Test Points (cont.)
• If the inequality is strict (lt or gt)
• Dont include values where x 0
• Dont include values where x is undefined.
• If the inequality is not strict ( or )
• Include values where x 0
• Dont include values where x is undefined.

138
Solving Inequalities Algebraically

• Example.
• Problem Solve the inequality x5 16x
• Answer

139
Key Points

• Solving Inequalities Algebraically