1. Copyright © 2014, 2010 Pearson Education, Inc. Chapter 2 Polynomials and Rational Functions Copyright © 2014, 2010 Pearson Education, Inc.

2. Section 2.1 Quadratic Functions 1. Graph a quadratic function in standard form. 2. Graph any quadratic function. 3. Solve problems modeled by quadratic functions. SECTION 1.1

3. Copyright © 2014, 2010 Pearson Education, Inc. 3 QUADRATIC FUNCTION A function of the form where a, b, and c, are real numbers with a ≠ 0, is called a quadratic function.

4. Copyright © 2014, 2010 Pearson Education, Inc. 4 THE STANDARD FORM OF A QUADRATIC FUNCTION The quadratic function is in standard form. The graph of f is a parabola with vertex (h, k). The parabola is symmetric with respect to the line x = h, called the axis of the parabola. If a > 0, the parabola opens up, and if a < 0, the parabola opens down.

5. Copyright © 2014, 2010 Pearson Education, Inc. 5 Find the standard form of the quadratic function whose graph has vertex (–3, 4) and passes through the point (–4, 7). Let y = f (x) be the quadratic function. Example: Writing the Equation of a Quadratic Function

6. Copyright © 2014, 2010 Pearson Education, Inc. 6 Example: Graphing a Quadratic Function in Standard Form Sketch the graph of f (x) = a(x – h) 2 + k Step 1 The graph is a parabola because it has the form f(x) = a(x – h) 2 + k Identify a, h, and k. Sketch the graph of 1. The graph of f(x) = –3(x + 2) 2 + 12 is the parabola: a = –3, h= – 2, and k = 12.

7. Copyright © 2014, 2010 Pearson Education, Inc. 7 Example: Graphing a Quadratic Function in Standard Form Step 2 Determine how the parabola opens. If a > 0, the parabola opens up. If a < 0, the parabola opens down. Step 3 Find the vertex. The vertex is (h, k). If a > 0 (or a < 0), the function f has a minimum (or a maximum) value k at x = h. 2. Since a = –3, a < 0, the parabola opens down. 3. The vertex (h,k) = (–2, 12). Since the parabola opens down, the function f has a maximum value of 12 at x = –2.

8. Copyright © 2014, 2010 Pearson Education, Inc. 8 Example: Graphing a Quadratic Function in Standard Form Step 4 Find the x- intercepts (if any). Set f (x) = 0 and solving the equation a(x – h) 2 + k = 0 for x. If the solutions are real numbers, they are the x-intercepts. If not, the parabola lies above the x- axis (when a > 0) or below the x-axis (when a < 0). 4.

9. Copyright © 2014, 2010 Pearson Education, Inc. 9 Example: Graphing a Quadratic Function in Standard Form Step 5 Find the y-intercept. Replace x with 0. Then f (0) = ah 2 + k is the y-intercept. 5.

10. Copyright © 2014, 2010 Pearson Education, Inc. 10 Example: Graphing a Quadratic Function in Standard Form Step 6 Sketch the graph. Plot the points found in Steps 3–5 and join them by a parabola. Show the axis x = h of the parabola by drawing a dashed line. 6. axis: x = –2

11. Copyright © 2014, 2010 Pearson Education, Inc. 11 Example: Graphing a Quadratic Function Graph f(x) = ax 2 + bx + c, a ≠ 0. Step 1 Identify a, b, and c. Sketch the graph of 1. In the equation y = f(x) = 2x 2 + 8x – 10, a = 2, b = 8, and c = – 10.

12. Copyright © 2014, 2010 Pearson Education, Inc. 12 Example: Graphing a Quadratic Function Step 2 Determine how the parabola opens. If a > 0, the parabola opens up. If a < 0, the parabola opens down. Step 3 Find the vertex. 2. Since a = 2 > 0, the parabola opens up. 3.

13. Copyright © 2014, 2010 Pearson Education, Inc. 13 Example: Graphing a Quadratic Function Step 4 Find the x- intercepts (if any). Set f (x) = 0 and solving the equation a(x – h) 2 + k = 0 for x. If the solutions are real numbers, they are the x-intercepts. If not, the parabola lies above the x- axis (when a > 0) or below the x-axis (when a < 0). 4.

14. Copyright © 2014, 2010 Pearson Education, Inc. 14 Example: Graphing a Quadratic Function Step 5 Find the y-intercept. Let x = 0. The result, f (0) = c, is the y-intercept. 5. Let x = 0.

15. Copyright © 2014, 2010 Pearson Education, Inc. 15 Example: Graphing a Quadratic Function Step 6 The parabola is symmetric with respect to its axis, Use this symmetry to find additional points. 6. Axis of symmetry is x = –2. The symmetric image of (0, –10) with respect to the axis x = –2 is (–4, –10).

16. Copyright © 2014, 2010 Pearson Education, Inc. 16 Example: Graphing a Quadratic Function Step 7 Draw a parabola through the points found in Steps 3–6. 7. Sketch the parabola passing through the points found in Steps 3–6.

17. Copyright © 2014, 2010 Pearson Education, Inc. 17 The graph of f (x) = –2x 2 +8x – 5 is shown. a.Find the domain and range of f. b.Solve the inequality –2x 2 +8x – 5 > 0. b.The graph is above the x- axis in the interval a.The domain is (–∞, ∞). The range is (–∞, 3]. Example: Identifying the Characteristics of a Quadratic Function from Its Graph

18. Copyright © 2014, 2010 Pearson Education, Inc. 18 Section 2.2 Polynomial Functions 1. Learn properties of the graphs of polynomial functions. 2. Determine the end behavior of polynomial functions. 3. Find the zeros of a polynomial function by factoring. 4. Identify the relationship between degrees, real zeros, and turning points. 5. Graph polynomial functions. SECTION 1.1

19. Copyright © 2014, 2010 Pearson Education, Inc. 19 Definitions A polynomial function of degree n is a function of the form where n is a nonnegative integer and the coefficients a n, a n–1, …, a 2, a 1, a 0 are real numbers with a n ≠ 0.

20. Copyright © 2014, 2010 Pearson Education, Inc. 20 Definitions The term a n x n is called the leading term. The number a n is called the leading coefficient, and a 0 is the constant term. A constant function f (x) = a, (a ≠ 0) which may be written as f (x) = ax 0, is a polynomial of degree 0.

21. Copyright © 2014, 2010 Pearson Education, Inc. 21 Definitions The zero function f (x) = 0 has no degree assigned to it. Polynomials of degree 3, 4, and 5 are called cubic, quartic, and quintic polynomials.

22. Copyright © 2014, 2010 Pearson Education, Inc. 22 Definitions The expression is a polynomial, the function f defined by is a polynomial function, and the equation is a polynomial equation.

23. Copyright © 2014, 2010 Pearson Education, Inc. 23 COMMON PROPERTIES OF POLYNOMIAL FUNCTIONS 1.The domain of a polynomial function is the set of all real numbers.

24. Copyright © 2014, 2010 Pearson Education, Inc. 24 2.The graph of a polynomial function is a continuous curve.

25. Copyright © 2014, 2010 Pearson Education, Inc. 25 3.The graph of a polynomial function is a smooth curve.

26. Copyright © 2014, 2010 Pearson Education, Inc. 26 State which functions are polynomial functions. For each polynomial function, find its degree, the leading term, and the leading coefficient. a. f (x) = 5x 4 – 2x + 7 b. g(x) = 7x 2 – x + 1, 1  x  5 a. f (x) is a polynomial function. Its degree is 4, the leading term is 5x 4, and the leading coefficient is 5. b. g(x) is not a polynomial function because its domain is not (– ,  ). Example: Polynomial Functions

27. Copyright © 2014, 2010 Pearson Education, Inc. 27 POWER FUNCTION A function of the form is called a power function of degree n, where a is a nonzero real number and n is a positive integer.

28. Copyright © 2014, 2010 Pearson Education, Inc. 28 POWER FUNCTIONS OF EVEN DEGREE The graph is symmetric with respect to the y-axis. The graph of y = x n (n is even) is similar to the graph of y = x 2. Let If n is even, then Then

29. Copyright © 2014, 2010 Pearson Education, Inc. 29 POWER FUNCTIONS OF ODD DEGREE The graph is symmetric with respect to the origin. The graph of y = x n (n is odd) is similar to the graph of y = x 3. LetIf n is odd, then Then

30. Copyright © 2014, 2010 Pearson Education, Inc. 30 END BEHAVIOR OF POLYNOMIAL FUNCTIONS Case 1 a > 0 The graph rises to the left and right, similar to y = x 2.

31. Copyright © 2014, 2010 Pearson Education, Inc. 31 END BEHAVIOR OF POLYNOMIAL FUNCTIONS Case 2 a < 0 The graph falls to the left and right, similar to y = –x 2.

32. Copyright © 2014, 2010 Pearson Education, Inc. 32 END BEHAVIOR OF POLYNOMIAL FUNCTIONS Case 3 a > 0 The graph rises to the right and falls to the left, similar to y = x 3.

33. Copyright © 2014, 2010 Pearson Education, Inc. 33 END BEHAVIOR OF POLYNOMIAL FUNCTIONS Case 4 a < 0 The graph rises to the left and falls to the right, similar to y = –x 3.

34. Copyright © 2014, 2010 Pearson Education, Inc. 34 Let be a polynomial function of degree 3. Show that when |x| is very large. When |x| is very large are close to 0. Therefore, Example: Understanding the End Behavior of a Polynomial Function

35. Copyright © 2014, 2010 Pearson Education, Inc. 35 THE LEADING–TERM TEST Its leading term is a n x n. The behavior of the graph of f as x → ∞ or as x → –∞ is similar to one of the following four graphs and is described as shown in each case. The middle portion of each graph, indicated by the dashed lines, is not determined by this test. Let be a polynomial function.

36. Copyright © 2014, 2010 Pearson Education, Inc. 36 Case 1 a n > 0 THE LEADING – TERM TEST

37. Copyright © 2014, 2010 Pearson Education, Inc. 37 Case 2 a n < 0 THE LEADING – TERM TEST

38. Copyright © 2014, 2010 Pearson Education, Inc. 38 Case 3 a n > 0 THE LEADING – TERM TEST

39. Copyright © 2014, 2010 Pearson Education, Inc. 39 Case 4 a n < 0 THE LEADING – TERM TEST

40. Copyright © 2014, 2010 Pearson Education, Inc. 40 Use the leading-term test to determine the end behavior of the graph of Here n = 3 (odd) and a n = –2 < 0. Thus, Case 4 applies. The graph of f (x) rises to the left and falls to the right. This behavior is described as y → ∞ as x → –∞ and y → –∞ as x → ∞. Example: Using the Leading-Term Test

41. Copyright © 2014, 2010 Pearson Education, Inc. 41 REAL ZEROS OF POLYNOMIAL FUNCTIONS 1.c is a zero of f. 2.c is a solution (or root) of the equation f (x) = 0. 3.c is an x-intercept of the graph of f. The point (c, 0) is on the graph of f. If f is a polynomial function and c is a real number, then the following statements are equivalent.

42. Copyright © 2014, 2010 Pearson Education, Inc. 42 Find all zeros of each polynomial function. Factor f (x) and then solve f (x) = 0. Example: Finding the Zeros of a Polynomial Function

43. Copyright © 2014, 2010 Pearson Education, Inc. 43 The only zero of g(x) is 2, since x 2 + 1 > 0 for all real numbers x. Example: Finding the Zeros of a Polynomial Function

44. Copyright © 2014, 2010 Pearson Education, Inc. 44 REAL ZEROS OF POLYNOMIAL FUNCTIONS A polynomial function of degree n with real coefficients has at most n real zeros.

45. Copyright © 2014, 2010 Pearson Education, Inc. 45 Find the number of distinct real zeros of the following polynomial functions of degree 3. a. Solve f (x) = 0. f (x) has three real zeros: –2, 1, and 3. Example: Finding the Number of Real Zeros

46. Copyright © 2014, 2010 Pearson Education, Inc. 46 h(x) has two distinct real zeros: –1 and 3. g(x) has only one real zero: –1. Example: Finding the Number of Real Zeros

47. Copyright © 2014, 2010 Pearson Education, Inc. 47 MULTIPLICITY OF A ZERO If c is a zero of a polynomial function f (x) and the corresponding factor (x – c) occurs exactly m times when f (x) is factored, then c is called a zero of multiplicity m. 1.If m is odd, the graph of f crosses the x-axis at x = c. 2.If m is even, the graph of f touches but does not cross the x-axis at x = c.

48. Copyright © 2014, 2010 Pearson Education, Inc. 48 MULTIPLICITY OF A ZERO

49. Copyright © 2014, 2010 Pearson Education, Inc. 49 MULTIPLICITY OF A ZERO

50. Copyright © 2014, 2010 Pearson Education, Inc. 50 Find the zeros of the polynomial function f (x) = x 2 (x + 1)(x – 2), and give the multiplicity of each zero. f (x) is already in factored form. f (x) = x 2 (x + 1)(x – 2) = 0 x 2 = 0, or x + 1 = 0, or x – 2 = 0 x = 0 or x = –1 or x = 2 The zero x = 0 has multiplicity 2, while each of the zeros –1 and 2 have multiplicity 1. Example: Finding the Zeros and Their Multiplicities

51. Copyright © 2014, 2010 Pearson Education, Inc. 51 TURNING POINTS A local (or relative) maximum value of f is higher than any nearby point on the graph. A local (or relative) minimum value of f is lower than any nearby point on the graph. The graph points corresponding to the local maximum and local minimum values are called turning points. At each turning point the graph changes from increasing to decreasing or vice versa.

52. Copyright © 2014, 2010 Pearson Education, Inc. 52 TURNING POINTS The graph of f has turning points at (–1, 12) and at (2, –15).

53. Copyright © 2014, 2010 Pearson Education, Inc. 53 NUMBER OF TURNING POINTS If f (x) is a polynomial of degree n, then the graph of f has at most (n – 1) turning points.

54. Copyright © 2014, 2010 Pearson Education, Inc. 54 Use a graphing calculator and the window –10  x  10; –30  y  30 to find the number of turning points of the graph of each polynomial. Example: Finding the Number of Turning Points

55. Copyright © 2014, 2010 Pearson Education, Inc. 55 f has three total turning points; two local minimum and one local maximum. Example: Finding the Number of Turning Points

56. Copyright © 2014, 2010 Pearson Education, Inc. 56 g has two total turning points; one local maximum and one local minimum. Example: Finding the Number of Turning Points

57. Copyright © 2014, 2010 Pearson Education, Inc. 57 h has no turning points, it is increasing on the interval (–∞, ∞). Example: Finding the Number of Turning Points

58. Copyright © 2014, 2010 Pearson Education, Inc. 58 Example: Graphing a Polynomial Function Sketch the graph of a polynomial function Step 1 Determine the end behavior. Apply the leading-term test. Sketch the graph of 1. Degree = 3 Leading coefficient = –1 End behavior shown.

59. Copyright © 2014, 2010 Pearson Education, Inc. 59 Example: Graphing a Polynomial Function Step 2Find the zeros of the polynomial function. Set f (x) = 0 and solve. The zeros give the x-intercepts. 2. Each zero has multiplicity 1, the graph crosses the x-axis at each zero.

60. Copyright © 2014, 2010 Pearson Education, Inc. 60 Example: Graphing a Polynomial Function Step 3 Find the y-intercept by computing f (0). Step 4 Use symmetry to check whether the function is odd, even, or neither. 3. The y-intercept is f(0) = 16. the graph passes through the point (0,16). 4. There is no symmetry in the y-axis nor with respect to the origin.

61. Copyright © 2014, 2010 Pearson Education, Inc. 61 Example: Graphing a Polynomial Function Step 5 Determine the sign of f(x) by using arbitrarily chosen “test numbers” in the intervals defined by the x-intercepts. Find the intervals on which the graph lies above or below the x- axis. 5.

62. Copyright © 2014, 2010 Pearson Education, Inc. 62 Example: Graphing a Polynomial Function Step 6 Draw the graph. Use the fact that the number of turning points is less than the degree of the polynomial to check whether the graph is drawn correctly. 6. Draw the graph. The number of turning points is 2, which is less than 3, the degree of f.

63. Copyright © 2014, 2010 Pearson Education, Inc. 63 Section 2.3 Dividing Polynomials and the Rational Zeros Test 1. Learn the Division Algorithm. 2. Use the Remainder and Factor Theorems. 3. Use the Rational Zeros Test. SECTION 1.1

64. Copyright © 2014, 2010 Pearson Education, Inc. 64 POLYNOMIAL FACTOR A polynomial D(x) is a factor of a polynomial F(x) if there is a polynomial Q(x) such that F(x) = D(x) ∙ Q(x).

65. Copyright © 2014, 2010 Pearson Education, Inc. 65 THE DIVISION ALGORITHM If a polynomial F(x) is divided by a polynomial D(x), with D(x) ≠ 0, there are unique polynomials Q(x) and R(x) such that Either R(x) is the zero polynomial, or the degree of R(x) is less than the degree of D(x).

66. Copyright © 2014, 2010 Pearson Education, Inc. 66 Use long division and synthetic division to find the quotient and remainder when 2x 4 + x 3 − 16x 2 + 18 is divided by x + 2. We will start by performing long division. Example: Using Long Division and Synthetic Division

67. Copyright © 2014, 2010 Pearson Education, Inc. 67 Example: Using Long Division and Synthetic Division

68. Copyright © 2014, 2010 Pearson Education, Inc. 68 quotient: 2x 3 – 3x 2 – 10x + 20 remainder: – 22 The result is Example: Using Long Division and Synthetic Division

69. Copyright © 2014, 2010 Pearson Education, Inc. 69 THE REMAINDER THEOREM If a polynomial F(x) is divided by x – a, then the remainder R is given by

70. Copyright © 2014, 2010 Pearson Education, Inc. 70 Find the remainder when the polynomial By the Remainder Theorem, F(1) is the remainder. The remainder is –2. Example: Using the Remainder Theorem

71. Copyright © 2014, 2010 Pearson Education, Inc. 71 Let One way is to evaluate f (x) when x = –3. Another way is to use synthetic division. Either method yields a value of 6. Example: Using the Remainder Theorem

72. Copyright © 2014, 2010 Pearson Education, Inc. 72 THE FACTOR THEOREM A polynomial F(x) has (x – a) as a factor if and only if F(a) = 0.

73. Copyright © 2014, 2010 Pearson Education, Inc. 73 Given that 2 is a zero of the function solve the polynomial equation Since 2 is a zero of f (x), f (2) = 0 and (x – 2) is a factor of f (x). Perform synthetic division by 2. Example: Using the Factor Theorem

74. Copyright © 2014, 2010 Pearson Education, Inc. 74 Since the remainder is 0, To find other zeros, solve the depressed equation. Example: Using the Factor Theorem

75. Copyright © 2014, 2010 Pearson Education, Inc. 75 Including the original zero of 2, the solution set is Example: Using the Factor Theorem

76. Copyright © 2014, 2010 Pearson Education, Inc. 76 THE RATIONAL ZEROS TEST 1. p is a factor of the constant term a 0 ; 2. q is a factor of the leading coefficient a n. is a polynomial function with integer coefficients (a n ≠ 0, a 0 ≠ 0) and is a rational number in lowest terms that is a zero of F(x), then

77. Copyright © 2014, 2010 Pearson Education, Inc. 77 Find all the rational zeros of List all possible zeros Example: Using the Rational Zeros Test

78. Copyright © 2014, 2010 Pearson Education, Inc. 78 Begin testing with 1. if it is not a rational zero, then try another possible zero. The remainder of 0 tells us that (x – 1) is a factor of F(x). The other factor is 2x 2 + 7x + 3. To find the other zeros, solve 2x 2 + 7x + 3 = 0. Example: Using the Rational Zeros Test

79. Copyright © 2014, 2010 Pearson Education, Inc. 79 The solution set is The rational zeros of F are Example: Using the Rational Zeros Test

80. Copyright © 2014, 2010 Pearson Education, Inc. 80 Section 2.4 Zeros of a Polynomial Function 1. Learn basic facts about the complex zeros of polynomials. 2. Use the Conjugate Pairs Theorem to find zeros of polynomials. SECTION 1.1

81. Copyright © 2014, 2010 Pearson Education, Inc. 81 If we extend our number system to allow the coefficients of polynomials and variables to represent complex numbers, we call the polynomial a complex polynomial. If P(z) = 0 for a complex number z we say that z is a zero or a complex zero of P(x). In the complex number system, every nth-degree polynomial equation has exactly n roots and every nth- degree polynomial can be factored into exactly n linear factors. Definitions

82. Copyright © 2014, 2010 Pearson Education, Inc. 82 FUNDAMENTAL THEOREM OF ALGEBRA Every polynomial with complex coefficients a n, a n – 1, …, a 1, a 0 has at least one complex zero.

83. Copyright © 2014, 2010 Pearson Education, Inc. 83 FACTORIZATION THEOREM FOR POLYNOMIALS If P(x) is a complex polynomial of degree n ≥ 1, it can be factored into n (not necessarily distinct) linear factors of the form where a, r 1, r 2, …, r n are complex numbers.

84. Copyright © 2014, 2010 Pearson Education, Inc. 84 Find a polynomial P(x) of degree 4 with a leading coefficient of 2 and zeros –1, 3, i, and –i. Write P(x) a. Since P(x) has degree 4, we write a.in completely factored form; b.by expanding the product found in part a. Example: Constructing a Polynomial Whose Zeros Are Given

85. Copyright © 2014, 2010 Pearson Education, Inc. 85 b. Expand the product found in part a. Example: Constructing a Polynomial Whose Zeros Are Given

86. Copyright © 2014, 2010 Pearson Education, Inc. 86 CONJUGATE PAIRS THEOREM If P(x) is a polynomial function whose coefficients are real numbers and if z = a + bi is a zero of P, then its conjugate, is also a zero of P.

87. Copyright © 2014, 2010 Pearson Education, Inc. 87 ODD – DEGREE POLYNOMIALS WITH REAL ZEROS Any polynomial P(x) of odd degree with real coefficients must have at least one real zero.

88. Copyright © 2014, 2010 Pearson Education, Inc. 88 A polynomial P(x) of degree 9 with real coefficients has the following zeros: 2, of multiplicity 3; 4 + 5i, of multiplicity 2; and 3 – 7i. Write all nine zeros of P(x). Since complex zeros occur in conjugate pairs, the conjugate 4 – 5i of 4 + 5i is a zero of multiplicity 2, and the conjugate 3 + 7i of 3 – 7i is a zero of P(x). The nine zeros of P(x) are: 2, 2, 2, 4 + 5i, 4 – 5i, 4 + 5i, 4 – 5i, 3 + 7i, 3 – 7i Example: Using the Conjugate Pairs Theorem

89. Copyright © 2014, 2010 Pearson Education, Inc. 89 FACTORIZATION THEOREM FOR A POLYNOMIAL WITH REAL COEFFICIENTS Every polynomial with real coefficients can be uniquely factored over the real numbers as a product of linear factors and/or irreducible quadratic factors.

90. Copyright © 2014, 2010 Pearson Education, Inc. 90 Given that 2 – i is a zero of find the remaining zeros. The conjugate of 2 – i, 2 + i is also a zero. So P(x) has linear factors: Example: Finding the Complex Zeros of a Polynomial

91. Copyright © 2014, 2010 Pearson Education, Inc. 91 Divide P(x) by x 2 – 4x + 5. Example: Finding the Complex Zeros of a Polynomial

92. Copyright © 2014, 2010 Pearson Education, Inc. 92 Therefore The zeros of P(x) are 1 (of multiplicity 2), 2 – i, and 2 + i. Example: Finding the Complex Zeros of a Polynomial

93. Copyright © 2014, 2010 Pearson Education, Inc. 93 Find all zeros of the polynomial P(x) = x 4 – x 3 + 7x 2 – 9x – 18. Possible zeros are: ±1, ±2, ±3, ±6, ±9, ±18 Use synthetic division to find that 2 is a zero. (x – 2) is a factor of P(x). Solve Example: Finding the Zeros of a Polynomial

94. Copyright © 2014, 2010 Pearson Education, Inc. 94 The four zeros of P(x) are –1, 2, –3i, and 3i. Example: Finding the Zeros of a Polynomial

95. Copyright © 2014, 2010 Pearson Education, Inc. 95 Section 2.5 Rational Functions 1. Define a rational function. 2. Define vertical and horizontal asymptotes. 3. Graph translations of 4. Find vertical and horizontal asymptotes (if any). 5. Graph rational functions. 6. Graph rational functions with oblique asymptotes. 7. Graph a revenue curve. SECTION 1.1

96. Copyright © 2014, 2010 Pearson Education, Inc. 96 RATIONAL FUNCTION A function f that can be expressed in the form where the numerator N(x) and the denominator D(x) are polynomials and D(x) is not the zero polynomial, is called a rational function. The domain of f consists of all real numbers for which D(x) ≠ 0.

97. Copyright © 2014, 2010 Pearson Education, Inc. 97 Find the domain of each rational function. a.The domain of f (x) is the set of all real numbers for which x – 1 ≠ 0; that is, x ≠ 1. In interval notation: Example: Finding the Domain of a Rational Function

98. Copyright © 2014, 2010 Pearson Education, Inc. 98 b.Find the values of x for which the denominator x 2 – 6x + 8 = 0, then exclude those values from the domain. The domain of g (x) is the set of all real numbers such that x ≠ 2 and x ≠ 4. In interval notation: Example: Finding the Domain of a Rational Function

99. Copyright © 2014, 2010 Pearson Education, Inc. 99 Example: Finding the Domain of a Rational Function c.The domain of is the set of all real numbers for which x – 2 ≠ 0; that is, x ≠ 2. The domain of g (x) is the set of all real numbers such that x ≠ 2. In interval notation:

100. Copyright © 2014, 2010 Pearson Education, Inc. 100 VERTICAL ASYMPTOTES The line with equation x = a is called a vertical asymptote of the graph of a function f if

101. Copyright © 2014, 2010 Pearson Education, Inc. 101 VERTICAL ASYMPTOTES

102. Copyright © 2014, 2010 Pearson Education, Inc. 102 VERTICAL ASYMPTOTES

103. Copyright © 2014, 2010 Pearson Education, Inc. 103 LOCATING VERTICAL ASYMPTOTES OF RATIONAL FUNCTIONS Ifis a rational function, where the N(x) and D(x) do not have a common factor and a is a real zero of D(x), then the line with equation x = a is a vertical asymptote of the graph of f.

104. Copyright © 2014, 2010 Pearson Education, Inc. 104 Find all vertical asymptotes of the graph of each rational function. a. No common factors, zero of the denominator is x = 1. The line with equation x = 1 is a vertical asymptote of f (x). Example: Finding Vertical Asymptotes

105. Copyright © 2014, 2010 Pearson Education, Inc. 105 b.No common factors. Factoring x 2 – 9 = (x + 3)(x – 3), we see the zeros of the denominator are –3 and 3. The lines with equations x = – 3 and x = 3 are the two vertical asymptotes of g (x). c.The denominator x 2 + 1 has no real zeros. Hence, the graph of the rational function h (x) has no vertical asymptotes. Example: Finding Vertical Asymptotes

106. Copyright © 2014, 2010 Pearson Education, Inc. 106 Find all vertical asymptotes of the graph of each rational function. The graph is the line with equation y = x + 3, with a gap (hole) corresponding to x = 3. Example: Rational Function Whose Graph Has a Hole

107. Copyright © 2014, 2010 Pearson Education, Inc. 107 Example: Rational Function Whose Graph Has a Hole

108. Copyright © 2014, 2010 Pearson Education, Inc. 108 The graph has a hole at x = –2. However, the graph of g(x) also has a vertical asymptote at x = 2. Example: Rational Function Whose Graph Has a Hole

109. Copyright © 2014, 2010 Pearson Education, Inc. 109 Example: Rational Function Whose Graph Has a Hole

110. Copyright © 2014, 2010 Pearson Education, Inc. 110 HORIZONTAL ASYMPTOTES The line with equation y = k is called a horizontal asymptote of the graph of a function f if

111. Copyright © 2014, 2010 Pearson Education, Inc. 111 RULES FOR LOCATING HORIZONTAL ASYMPTOTES Let f be a rational function given by To find whether the graph of f has one horizontal asymptote or no horizontal asymptote, we compare the degree of the numerator, n, with that of the denominator, m:

112. Copyright © 2014, 2010 Pearson Education, Inc. 112 1.If n < m, then the x-axis (y = 0) is the horizontal asymptote. 2.If n = m, then the line with equation is the horizontal asymptote. 3.If n > m, then the graph of f has no horizontal asymptote. RULES FOR LOCATING HORIZONTAL ASYMPTOTES

113. Copyright © 2014, 2010 Pearson Education, Inc. 113 Find the horizontal asymptote (if any) of the graph of each rational function. a.Numerator and denominator have degree 1. is the horizontal asymptote. Example: Finding the Horizontal Asymptote

114. Copyright © 2014, 2010 Pearson Education, Inc. 114 degree of denominator > degree of numerator y = 0 (the x-axis) is the horizontal asymptote. degree of numerator > degree of denominator The graph has no horizontal asymptote. Example: Finding the Horizontal Asymptote

115. Copyright © 2014, 2010 Pearson Education, Inc. 115 Example: Graphing a Rational Function Graph where f(x) is in lowest terms. Step 1 Find the intercepts. Since f is in lowest terms, the x-intercepts are found by solving the equation N(x) = 0. The y-intercept is, if there is one, is f (0). Sketch the graph of 1. The x-intercepts are  1 and 1, so the graph passes through (  1, 0) and (1, 0).

116. Copyright © 2014, 2010 Pearson Education, Inc. 116 Example: Graphing a Rational Function Step 1 Find the intercepts. Since f is in lowest terms, the x-intercepts are found by solving the equation N(x) = 0. The y-intercept is, if there is one, is f (0). 1. The y-intercept is, so the graph of f passes through the point (0, ).

117. Copyright © 2014, 2010 Pearson Education, Inc. 117 Example: Graphing a Rational Function Step 2 Find the vertical asymptotes (if any). Solve D(x) = 0 to find the vertical asymptotes of the graph. Sketch the vertical asymptotes. 2. The vertical asymptotes are x = 3 and x = –3.

118. Copyright © 2014, 2010 Pearson Education, Inc. 118 Example: Graphing a Rational Function Step 3 Find the horizontal asymptotes (if any). Use the rules from the previous slide. 3. Since n = m = 2 the horizontal asymptote is

119. Copyright © 2014, 2010 Pearson Education, Inc. 119 Example: Graphing a Rational Function Step 4 Test for symmetry. If f (–x) = f (x), then f is symmetric with respect to the y-axis. If f (–x) = –f (x), then f is symmetric with respect to the origin. 4. The graph of f is symmetric in the y-axis. This is only one symmetry.

120. Copyright © 2014, 2010 Pearson Education, Inc. 120 Example: Graphing a Rational Function Step 5 Locate the graph relative to the horizontal asymptote (if any). Use the sign graphs and test numbers associated with the zeros of N(x) and D(x), to determine where the graph of f is above the x-axis and where it is below the x-axis. 5. R(x) = 16 has no zeros and D(x) has zeros –3 and 3. These zeros divide the x- axis into three intervals. We choose test points –4, 0, and 4 to test the sign of

121. Copyright © 2014, 2010 Pearson Education, Inc. 121 Example: Graphing a Rational Function Step 6 Sketch the graph. Plot some points and graph the asymptotes found in Steps 1-5; use symmetry to sketch the graph of f. 6.

122. Copyright © 2014, 2010 Pearson Education, Inc. 122 Sketch the graph of Step 1Since x 2 + 2 > 0, no x-intercepts. ; y-intercept is –1. Step 2Solve (x + 2)(x – 1) = 0; x = –2, x = 1 Vertical asymptotes are x = –2 and x = 1. Example: Graphing a Rational Function

123. Copyright © 2014, 2010 Pearson Education, Inc. 123 Step 3 degree of den = degree of num y = 1 is the horizontal asymptote Step 4Symmetry. None Step 5The zeros of the denominator –2 and 1 yield the following figure: Example: Graphing a Rational Function

124. Copyright © 2014, 2010 Pearson Education, Inc. 124 Example: Graphing a Rational Function

125. Copyright © 2014, 2010 Pearson Education, Inc. 125 Step 6Sketch the graph. Example: Graphing a Rational Function

126. Copyright © 2014, 2010 Pearson Education, Inc. 126 Sketch a graph of Step 1Since f (0) = 0 and setting f (x) = 0 yields 0, x-intercept and y-intercept are 0. Step 2Because x 2 +1 > 0 for all x, the domain is the set of all real numbers. Since there are no zeros for the denominator, there are no vertical asymptotes. Example: Graphing a Rational Function

127. Copyright © 2014, 2010 Pearson Education, Inc. 127 Step 3degree of den = degree of num y = 1 is the horizontal asymptote. Step 4Symmetry. Symmetric with respect to the y-axis. Step 5The graph is always above the x-axis, except at x = 0. Example: Graphing a Rational Function

128. Copyright © 2014, 2010 Pearson Education, Inc. 128 Step 6Sketch the graph. Example: Graphing a Rational Function

129. Copyright © 2014, 2010 Pearson Education, Inc. 129 OBLIQUE ASUMPTOTES Suppose is greater than the degree of D(x). Then and the degree of N(x) Thus, as That is, the graph of f approaches the graph of the oblique asymptote defined by Q(x).

130. Copyright © 2014, 2010 Pearson Education, Inc. 130 Sketch the graph of Step 2Solve x + 1 = 0; x = –1; domain is set of all real numbers except –1. Vertical asymptote is x = –1. ; y-intercept: –4. Step 1Solve x 2 – 4 = 0; x-intercepts: –2, 2 Example: Graphing a Rational Function with an Oblique Asymptote

131. Copyright © 2014, 2010 Pearson Education, Inc. 131 y = x – 1 is an oblique asymptote. Step 3 degree of numerator > degree of denominator Step 4Symmetry. None Step 5Use the intervals determined by the zeros of the denominator. Example: Graphing a Rational Function with an Oblique Asymptote

132. Copyright © 2014, 2010 Pearson Education, Inc. 132 Example: Graphing a Rational Function with an Oblique Asymptote

133. Copyright © 2014, 2010 Pearson Education, Inc. 133 Step 6Sketch the graph. Example: Graphing a Rational Function with an Oblique Asymptote