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- 1. Graphs of Factorable Polynomials
- 2. Graphs of Factorable Polynomials Following are some of the basic shapes of graphs that we encounter often. The dotted tangent line is for reference. Practice drawing them a few times.
- 3. Graphs of Factorable Polynomials We start with the graphs of the polynomials y = ±xN.
- 4. Graphs of Factorable Polynomials We start with the graphs of the polynomials y = ±xN. The graphs y = xeven y = x2
- 5. Graphs of Factorable Polynomials We start with the graphs of the polynomials y = ±xN. The graphs y = xeven y = x4 y = x2 (-1, 1) (1, 1)
- 6. Graphs of Factorable Polynomials We start with the graphs of the polynomials y = ±xN. The graphs y = xeven y = x6 y = x4 y = x2 (-1, 1) (1, 1)
- 7. Graphs of Factorable Polynomials We start with the graphs of the polynomials y = ±xN. The graphs y = xeven y = x6 y = x4 y = x2 The graphs y = –xeven y = -x2 y = -x4 y = -x6 (-1, 1) (1, 1) (-1,-1) (1,-1)
- 8. Graphs of Factorable Polynomials We start with the graphs of the polynomials y = ±xN. The graphs y = xeven y = x6 y = x4 y = x2 The graphs y = –xeven y = -x2 y = -x4 y = -x6 (-1, 1) (1, 1) (-1,-1) (1,-1) Plot these functions and zoom in on the region around x = –1 to x = 1. Note that the graphs in between the points (1, 1) and (–1,1) drop lower as the power increases. However the graphs switch positions as they pass to the right of (1, 1) or to the left of (–1,1). ( Why?)
- 9. Graphs of Factorable Polynomials We start with the graphs of the polynomials y = ±xN. The graphs y = xeven y = x6 y = x4 y = x2 (-1,-1) (1,-1) y = xE y = –xE (-1, 1) (1, 1) The graphs y = –xeven Graphs of even ordered powers y = ±xEven. y = -x2 y = -x4 y = -x6
- 10. Graphs of Factorable Polynomials The graphs y = xodd y = x3
- 11. Graphs of Factorable Polynomials The graphs y = xodd y = x5 y = x3 (1, 1) (-1, -1)
- 12. Graphs of Factorable Polynomials The graphs y = xodd y = x5 y = x3 y = x7 (1, 1) (-1, -1)
- 13. Graphs of Factorable Polynomials The graphs y = xodd y = x5 The graphs y = –xodd y = x7 y = -x3 y = x3 y = -x5 y = -x7 (1, 1) (-1, -1) (-1, 1) (1,-1)
- 14. Graphs of Factorable Polynomials The graphs y = xodd y = x5 The graphs y = –xodd y = x7 y = -x3 y = x3 y = -x5 y = -x7 y = xD y = –xD (1, 1) (-1, -1) (-1, 1) (1,-1) Graphs of odd– order powers y = ±xodd
- 15. Graphs of Factorable Polynomials Facts about the graphs of polynomials:
- 16. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves.
- 17. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves. • Polynomial curves are smooth (no corners).
- 18. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves. • Polynomial curves are smooth (no corners). • Let P(x) = anxn + lower degree terms.
- 19. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves. • Polynomial curves are smooth (no corners). • Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms.
- 20. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves. • Polynomial curves are smooth (no corners). • Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms. For x's such that | x | are large, the "lower degree terms" are negligible compared to anxn.
- 21. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves. • Polynomial curves are smooth (no corners). • Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms. For x's such that | x | are large, the "lower degree terms" are negligible compared to anxn. Hence, for x where |x| is "large", the graph of P(x) resembles the graph y = anxn.
- 22. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves. • Polynomial curves are smooth (no corners). • Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms. For x's such that | x | are large, the "lower degree terms" are negligible compared to anxn. Hence, for x where |x| is "large", the graph of P(x) resembles the graph y = anxn. This means there're four behaviors of polynomial-graphs to the far left or far right (as | x | becomes large).
- 23. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves. • Polynomial curves are smooth (no corners). • Let P(x) = anxn + lower degree terms. For large |x|, the leading term anxn dominates the lower degree terms. For x's such that | x | are large, the "lower degree terms" are negligible compared to anxn. Hence, for x where |x| is "large", the graph of P(x) rTehsise mmbelaens st hthee grrea'rpeh f oy u=r abnexhna. viors of polynomial-graphs to the far left or far right (as | x | becomes large). These behaviors are based on the sign the leading term anxn, and whether n is even or odd.
- 24. Graphs of Factorable Polynomials I. The "Arms" of Polynomial Graphs
- 25. Graphs of Factorable Polynomials I. The "Arms" of Polynomial Graphs y = +xeven + lower degree terms:
- 26. Graphs of Factorable Polynomials I. The "Arms" of Polynomial Graphs y = +xeven + lower degree terms: y = –xeven + lower degree terms:
- 27. Graphs of Factorable Polynomials I. The "Arms" of Polynomial Graphs y = +xeven + lower degree terms: y = –xeven + lower degree terms: y = +xodd + lower degree terms:
- 28. Graphs of Factorable Polynomials I. The "Arms" of Polynomial Graphs y = +xeven + lower degree terms: y = –xeven + lower degree terms: y = +xodd + lower degree terms: y = –xodd + lower degree terms:
- 29. Graphs of Factorable Polynomials For factorable polynomials, we use the sign-charts to sketch the central portion of the graphs.
- 30. Graphs of Factorable Polynomials For factorable polynomials, we use the sign-charts to sketch the central portion of the graphs. Recall that given a polynomial P(x), it's sign-chart is constructed in the following manner:
- 31. Graphs of Factorable Polynomials For factorable polynomials, we use the sign-charts to sketch the central portion of the graphs. Recall that given a polynomial P(x), it's sign-chart is constructed in the following manner: Construction of the sign-chart of polynomial P(x):
- 32. Graphs of Factorable Polynomials For factorable polynomials, we use the sign-charts to sketch the central portion of the graphs. Recall that given a polynomial P(x), it's sign-chart is constructed in the following manner: Construction of the sign-chart of polynomial P(x): I. Find the roots of P(x) and their order respectively.
- 33. Graphs of Factorable Polynomials For factorable polynomials, we use the sign-charts to sketch the central portion of the graphs. Recall that given a polynomial P(x), it's sign-chart is constructed in the following manner: Construction of the sign-chart of polynomial P(x): I. Find the roots of P(x) and their order respectively. II. Draw the real line, mark off the answers from I.
- 34. Graphs of Factorable Polynomials For factorable polynomials, we use the sign-charts to sketch the central portion of the graphs. Recall that given a polynomial P(x), it's sign-chart is constructed in the following manner: Construction of the sign-chart of polynomial P(x): I. Find the roots of P(x) and their order respectively. II. Draw the real line, mark off the answers from I. III. Sample a point for its sign, use the the orders of the roots to extend and fill in the signs.
- 35. Graphs of Factorable Polynomials For factorable polynomials, we use the sign-charts to sketch the central portion of the graphs. Recall that given a polynomial P(x), it's sign-chart is constructed in the following manner: Construction of the sign-chart of polynomial P(x): I. Find the roots of P(x) and their order respectively. II. Draw the real line, mark off the answers from I. III. Sample a point for its sign, use the the orders of the roots to extend and fill in the signs. (Reminder: Across odd-ordered root, sign changes Across even-ordered root, sign stays the same.)
- 36. Graphs of Factorable Polynomials Example A: Make the sign-chart of f(x) = x2 – 3x – 4 and graph y = f(x).
- 37. Graphs of Factorable Polynomials Example A: Make the sign-chart of f(x) = x2 – 3x – 4 and graph y = f(x). Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 so x = 4 and x= –1 are two roots of odd order.
- 38. Graphs of Factorable Polynomials Example A: Make the sign-chart of f(x) = x2 – 3x – 4 and graph y = f(x). Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 so x = 4 and x= –1 are two roots of odd order. The sign chart and the graph of y = f(x) are shown here. y=(x – 4)(x+1) x y
- 39. Graphs of Factorable Polynomials Example A: Make the sign-chart of f(x) = x2 – 3x – 4 and graph y = f(x). Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 so x = 4 and x= –1 are two roots of odd order. The sign chart and the graph of y = f(x) are shown here. y=(x – 4)(x+1) x y Note the sign-chart reflects the properties of the graph.
- 40. Graphs of Factorable Polynomials Example A: Make the sign-chart of f(x) = x2 – 3x – 4 and graph y = f(x). Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 so x = 4 and x= –1 are two roots of odd order. The sign chart and the graph of y = f(x) are shown here. y=(x – 4)(x+1) x y Note the sign-chart reflects the properties of the graph. I. The graph touches or crosses the x-axis at the roots.
- 41. Graphs of Factorable Polynomials Example A: Make the sign-chart of f(x) = x2 – 3x – 4 and graph y = f(x). Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 so x = 4 and x= –1 are two roots of odd order. The sign chart and the graph of y = f(x) are shown here. y=(x – 4)(x+1) x y Note the sign-chart reflects the properties of the graph. I. The graph touches or crosses the x-axis at the roots. II. The graph is above the x-axis where the sign is "+".
- 42. Graphs of Factorable Polynomials Example A: Make the sign-chart of f(x) = x2 – 3x – 4 and graph y = f(x). Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 so x = 4 and x= –1 are two roots of odd order. The sign chart and the graph of y = f(x) are shown here. y=(x – 4)(x+1) x y Note the sign-chart reflects the properties of the graph. I. The graph touches or crosses the x-axis at the roots. II. The graph is above the x-axis where the sign is "+". III. The graph is below the x-axis where the sign is "–".
- 43. Graphs of Factorable Polynomials II. The “Mid-Portions” of Polynomial Graphs
- 44. Graphs of Factorable Polynomials II. The “Mid-Portions” of Polynomial Graphs Graphs of an odd ordered root (x – r)odd at x = r.
- 45. Graphs of Factorable Polynomials II. The “Mid-Portions” of Polynomial Graphs Graphs of an odd ordered root (x – r)odd at x = r. + + r r order = 1 order = 1 y = (x – r)1 y = –(x – r)1 + r order = 3, 5, 7.. + r order = 3, 5, 7.. y = (x – r)3 or 5.. y = –(x – r)3 or 5..
- 46. Graphs of Factorable Polynomials Graphs of an even ordered root at (x – r)E at x= r. r + + order = 2, 4, 6 .. x=r order = 2, 4, 6 .. y = (x – r)2 or 4.. y = –(x – r)2 or 4.. If we know the roots of a factorable polynomial, then we may construct the central portion of the graph (the body) in the following manner using its sign chart. I. Draw the graph about each root using the information about the order of each root. II. Connect all the pieces together to form the graph.
- 47. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial,
- 48. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, + order=2 order=3
- 49. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown. + order=2 order=3
- 50. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown. Connect them to get the whole graph. + order=2 order=3
- 51. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown. Connect them to get the whole graph. + order=2 order=3
- 52. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown. Connect them to get the whole graph. Example B: Given P(x) = -x(x + 2)2(x – 3)2, identify the roots and their orders. Make the sign-chart. Sketch the graph about each root. Connect them to complete the graph. + order=2 order=3
- 53. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown. Connect them to get the whole graph. Example B: Given P(x) = -x(x + 2)2(x – 3)2, identify the roots and their orders. Make the sign-chart. Sketch the graph about each root. Connect them to complete the graph. The roots are x = 0 of order 1, + order=2 order=3
- 54. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown. Connect them to get the whole graph. Example B: Given P(x) = -x(x + 2)2(x – 3)2, identify the roots and their orders. Make the sign-chart. Sketch the graph about each root. Connect them to complete the graph. The roots are x = 0 of order 1, x = -2 of order 2, + order=2 order=3
- 55. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, the graphs around each root are as shown. Connect them to get the whole graph. Example B: Given P(x) = -x(x + 2)2(x – 3)2, identify the roots and their orders. Make the sign-chart. Sketch the graph about each root. Connect them to complete the graph. The roots are x = 0 of order 1, x = -2 of order 2, and x = 3 of order 2. + order=2 order=3
- 56. Graphs of Factorable Polynomials The sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = 3 order 2 x = 0 order 1 x = -2 order 2
- 57. Graphs of Factorable Polynomials The sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = 3 order 2 x = 0 order 1 x = -2 order 2 By the sign-chart and the order of each root, we draw the graph about each root.
- 58. Graphs of Factorable Polynomials The sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = 3 order 2 x = 0 order 1 x = -2 order 2 By the sign-chart and the order of each root, we draw the graph about each root.
- 59. Graphs of Factorable Polynomials The sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = 3 order 2 x = 0 order 1 x = -2 order 2 By the sign-chart and the order of each root, we draw the graph about each root. (Note for x = 0 of order 1, the graph approximates a line going through the point.)
- 60. Graphs of Factorable Polynomials The sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = 3 order 2 x = 0 order 1 x = -2 order 2 By the sign-chart and the order of each root, we draw the graph about each root. (Note for x = 0 of order 1, the graph approximates a line going through the point.) Connect all the pieces to get the graph of P(x).
- 61. Graphs of Factorable Polynomials The sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = -2 order 2 x = 0 order 1 x = 3 order 2 By the sign-chart and the order of each root, we draw the graph about each root. (Note for x = 0 of order 1, the graph approximates a line going through the point.) Connect all the pieces to get the graph of P(x).
- 62. Graphs of Factorable Polynomials Note the graph resembles y = -x5, it's leading term, when viewed at a distance.
- 63. Graphs of Factorable Polynomials Note the graph resembles y = -x5, it's leading term, when viewed at a distance. + + -2 0 3
- 64. Graphs of Rational Functions
- 65. Graphs of Rational Functions Recall that rational functions are functions of the form R(x) = P(x) where P(x) and Q(x) are Q(x) polynomials.
- 66. Graphs of Rational Functions Recall that rational functions are functions of the form R(x) = P(x) where P(x) and Q(x) are Q(x) polynomials. A rational function is factorable if both P(x) and Q(x) are factorable.
- 67. Graphs of Rational Functions Recall that rational functions are functions of the form R(x) = P(x) where P(x) and Q(x) are Q(x) polynomials. A rational function is factorable if both P(x) and Q(x) are factorable. In this section we study the graphs of reduced factorable rational functions.
- 68. Graphs of Rational Functions Recall that rational functions are functions of the form R(x) = P(x) where P(x) and Q(x) are Q(x) polynomials. A rational function is factorable if both P(x) and Q(x) are factorable. In this section we study the graphs of reduced factorable rational functions. The main principle of graphing these functions is the same as of polynomials. We analyze the behaviors and draw pieces of the graphs at important locations, then complete the graphs by connecting them. However, the behaviors of rational functions are more complicated due to the presence of the denominators.
- 69. x=0 Graphs of Rational Functions The graph of y = 1/x has an asymptote at x = 0 as shown here. – – – + + + Graph of y = 1/x Graph of y = ±1/xN for N = 1, 2, 3..
- 70. x=0 Graphs of Rational Functions The graph of y = 1/x has an asymptote at x = 0 as shown here. We also call vertical asymptotes “poles”. Therefore y = 1/x has a pole of order 1 at x = 0. – – – + + + Graph of y = 1/x Graph of y = ±1/xN for N = 1, 2, 3.. A pole of order 1
- 71. x=0 Graphs of Rational Functions The graph of y = 1/x has an asymptote at x = 0 as shown here. We also call vertical asymptotes “poles”. Therefore y = 1/x has a pole of order 1 at x = 0. – – – + + + Graph of y = 1/x Graph of y = ±1/xN for N = 1, 2, 3.. “Poles” are opposite of “roots”. A pole of order 1
- 72. x=0 Graphs of Rational Functions The graph of y = 1/x has an asymptote at x = 0 as shown here. We also call vertical asymptotes “poles”. Therefore y = 1/x has a pole of order 1 at x = 0. – – – + + + Graph of y = 1/x x=0 Graph of y = 1/x2 “Poles” are opposite of “roots”. The graph of y = 1/x2 has a pole of order 2 at x = 0. Its graph is shown here. + + + + + + Graph of y = ±1/xN for N = 1, 2, 3.. A pole of order 1 A pole of order 2
- 73. x=0 Graphs of Rational Functions The graph of y = 1/x has an asymptote at x = 0 as shown here. We also call vertical asymptotes “poles”. Therefore y = 1/x has a pole of order 1 at x = 0. – – – + + + Graph of y = 1/x x=0 Graph of y = 1/x2 “Poles” are opposite of “roots”. The graph of y = 1/x2 has a pole of order 2 at x = 0. Its graph is shown here. Just as “roots”, the even or odd orders of the “poles” determine the behaviors of the graphs. + + + + + + Graph of y = ±1/xN for N = 1, 2, 3.. A pole of order 1 A pole of order 2
- 74. Graphs of Rational Functions e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. + Graphs of odd ordered poles
- 75. Graphs of Rational Functions e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. + Graphs of + + Graphs of odd ordered poles even ordered poles e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..
- 76. Graphs of Rational Functions e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. + Graphs of + + Graphs of odd ordered poles even ordered poles e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,.. We are examining the “Mid–Portion” of the graphs of rational functions. We will look at the “Arms” of these graphs later.
- 77. Graphs of Rational Functions e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. + Graphs of + + Example A: Given the following information of roots, sign-chart and vertical asymptotes, draw the graph. root + + VA VA Graphs of odd ordered poles even ordered poles e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..
- 78. Graphs of Rational Functions Graphs of Graphs of odd ordered poles even ordered poles e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. + + + Example A: Given the following information of roots, sign-chart and vertical asymptotes, draw the graph. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,.. root + + Graph from right to left: VA VA
- 79. Graphs of Rational Functions Graphs of Graphs of odd ordered poles even ordered poles e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. + + + Example A: Given the following information of roots, sign-chart and vertical asymptotes, draw the graph. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,.. root + + Graph from right to left: VA VA
- 80. Graphs of Rational Functions Graphs of Graphs of odd ordered poles even ordered poles e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. + + + Example A: Given the following information of roots, sign-chart and vertical asymptotes, draw the graph. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,.. root + + Graph from right to left: VA VA
- 81. Graphs of Rational Functions Graphs of Graphs of odd ordered poles even ordered poles e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. + + + Example A: Given the following information of roots, sign-chart and vertical asymptotes, draw the graph. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,.. root + + Graph from right to left: VA VA
- 82. Graphs of Rational Functions Graphs of odd ordered poles even ordered poles + Graphs of e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. Example A: Given the following information of roots, sign-chart and vertical asymptotes, draw the graph. + + e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,.. root + + Graph from right to left: VA VA
- 83. Graphs of Rational Functions Graphs of odd ordered poles even ordered poles + Graphs of e.g. y = –1/x1 or 3,.. + e.g. y = 1/x1 or 3,.. Example A: Given the following information of roots, sign-chart and vertical asymptotes, draw the graph. + + e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,.. root + + Graph from right to left: VA VA
- 84. Graphs of Rational Functions Horizontal Asymptotes
- 85. Graphs of Rational Functions Horizontal Asymptotes For x's where | x | is large (i.e.. x is to the far right or far left on the x-axis),
- 86. Graphs of Rational Functions Horizontal Asymptotes For x's where | x | is large (i.e.. x is to the far right or far left on the x-axis), the graph of a rational function resembles the quotient of the leading terms of the numerator and the denominator.
- 87. Graphs of Rational Functions Horizontal Asymptotes For x's where | x | is large (i.e.. x is to the far right or far left on the x-axis), the graph of a rational function resembles the quotient of the leading terms of the numerator and the denominator. Specifically, if R(x) = AxN + lower degree terms BxK + lower degree terms
- 88. Graphs of Rational Functions Horizontal Asymptotes For x's where | x | is large (i.e.. x is to the far right or far left on the x-axis), the graph of a rational function resembles the quotient of the leading terms of the numerator and the denominator. Specifically, if R(x) = AxN + lower degree terms BxK + lower degree terms then for x's where | x | is large, the graph of R(x) resembles A x (quotient of the leading terms). N BxK
- 89. Graphs of Rational Functions Horizontal Asymptotes For x's where | x | is large (i.e.. x is to the far right or far left on the x-axis), the graph of a rational function resembles the quotient of the leading terms of the numerator and the denominator. Specifically, if R(x) = AxN + lower degree terms BxK + lower degree terms then for x's where | x | is large, the graph of R(x) resembles A x (quotient of the leading terms). N BxK The graph may or may not level off horizontally.
- 90. Graphs of Rational Functions Horizontal Asymptotes For x's where | x | is large (i.e.. x is to the far right or far left on the x-axis), the graph of a rational function resembles the quotient of the leading terms of the numerator and the denominator. Specifically, if R(x) = AxN + lower degree terms BxK + lower degree terms then for x's where | x | is large, the graph of R(x) resembles A x (quotient of the leading terms). N BxK The graph may or may not level off horizontally. If it does, then we have a horizontal asymptote (HA).
- 91. Graphs of Rational Functions We list all the possibilities of horizontal behavior below:
- 92. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior):
- 93. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms , the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B.
- 94. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms , the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B. I. If N > K,
- 95. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms , the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B. I. If N > K, then the graph of R(x) resembles the polynomial AxN-K/B.
- 96. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms , the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B. I. If N > K, then the graph of R(x) resembles the polynomial AxN-K/B. We write this as lim y = ±∞. x±∞
- 97. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms , the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B. I. If N > K, then the graph of R(x) resembles the polynomial AxN-K/B. We write this as lim y = ±∞. x±∞ II. If N = K,
- 98. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms , the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B. I. If N > K, then the graph of R(x) resembles the polynomial AxN-K/B. We write this as lim y = ±∞. x±∞ II. If N = K, then the graph of R(x) has y = A/B as a horizontal asymptote (HA).
- 99. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms , the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B. I. If N > K, then the graph of R(x) resembles the polynomial AxN-K/B. We write this as lim y = ±∞. II. If N = K, then the graph of R(x) has y = A/B as a horizontal asymptote (HA). It is noted as lim y = A/B. x±∞ x±∞
- 100. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms , the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B. I. If N > K, then the graph of R(x) resembles the polynomial AxN-K/B. We write this as lim y = ±∞. x±∞ II. If N = K, then the graph of R(x) has y = A/B as a horizontal asymptote (HA). It is noted as lim y = A/B. x±∞ III. If N < K,
- 101. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms , the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B. I. If N > K, then the graph of R(x) resembles the polynomial AxN-K/B. We write this as lim y = ±∞. x±∞ II. If N = K, then the graph of R(x) has y = A/B as a horizontal asymptote (HA). It is noted as lim y = A/B. x±∞ III. If N < K, then the graph of R(x) has y = 0 as a horizontal asymptote (HA) because N – K is negative.
- 102. Graphs of Rational Functions We list all the possibilities of horizontal behavior below: Theorem (Horizontal Behavior): Given that R(x) = AxN + lower degree terms BxK + lower degree terms the graph of R(x) as x goes to the far right (x ∞) and far left (x -∞) behaves similarly to AxN/BxK = AxN-K/B. I. If N > K, then the graph of R(x) resembles the polynomial AxN-K/B. II. If N = K, then the graph of R(x) has y = A/B as a horizontal asymptote (HA). It is noted as lim y = A/B. x±∞ III. If N < K, then the graph of R(x) has y = 0 as a horizontal asymptote (HA) because N – K is negative. It is noted as lim y = 0. x±∞ , We write this as lim y = ±∞. x±∞
- 103. Graphs of Rational Functions Steps for graphing a rational function R(x) = P ( x ) Q(x)
- 104. Graphs of Rational Functions Steps for graphing a rational function R(x) = I. (Roots) Find the roots of P(x) and their orders by solving P(x) = 0. P(x) Q(x)
- 105. Graphs of Rational Functions Steps for graphing a rational function R(x) = I. (Roots) Find the roots of P(x) and their orders by solving P(x) = 0. P(x) Q(x) II. (Poles or VA) Find the vertical asymptotes (VA) and their orders by solving Q(x) = 0.
- 106. Graphs of Rational Functions Steps for graphing a rational function R(x) = I. (Roots) Find the roots of P(x) and their orders by solving P(x) = 0. P(x) Q(x) II. (Poles or VA) Find the vertical asymptotes (VA) and their orders by solving Q(x) = 0. Steps I and II give the sign-chart,
- 107. Graphs of Rational Functions Steps for graphing a rational function R(x) = I. (Roots) Find the roots of P(x) and their orders by solving P(x) = 0. P(x) Q(x) II. (Poles or VA) Find the vertical asymptotes (VA) and their orders by solving Q(x) = 0. Steps I and II give the sign-chart, the shape of the graph around the roots, and the shape of the graph along the poles.
- 108. Graphs of Rational Functions Steps for graphing a rational function R(x) = I. (Roots) Find the roots of P(x) and their orders by solving P(x) = 0. P(x) Q(x) II. (Poles or VA) Find the vertical asymptotes (VA) and their orders by solving Q(x) = 0. Steps I and II give the sign-chart, the shape of the graph around the roots, and the shape of the graph along the poles. Steps I and II give the "mid-section“ of the graph.
- 109. Graphs of Rational Functions Steps for graphing a rational function R(x) = I. (Roots) Find the roots of P(x) and their orders by solving P(x) = 0. P(x) Q(x) II. (Poles or VA) Find the vertical asymptotes (VA) and their orders by solving Q(x) = 0. Steps I and II give the sign-chart, the shape of the graph around the roots, and the shape of the graph along the poles. Steps I and II give the "mid-section“ of the graph. III. (HA) Use the last theorem to determine the behavior of the graph to the right and left as x±∞.
- 110. Graphs of Rational Functions Steps for graphing a rational function R(x) = I. (Roots) Find the roots of P(x) and their orders by solving P(x) = 0. P(x) Q(x) II. (Poles or VA) Find the vertical asymptotes (VA) and their orders by solving Q(x) = 0. Steps I and II give the sign-chart, the shape of the graph around the roots, and the shape of the graph along the poles. Steps I and II give the "mid-section“ of the graph. III. (HA) Use the last theorem to determine the behavior of the graph to the right and left as x±∞. The horizontal asymptote exists only if the limit exists.
- 111. Graphs of Rational Functions Steps for graphing a rational function R(x) = I. (Roots) Find the roots of P(x) and their orders by solving P(x) = 0. P(x) Q(x) II. (Poles or VA) Find the vertical asymptotes (VA) and their orders by solving Q(x) = 0. Steps I and II give the sign-chart, the shape of the graph around the roots, and the shape of the graph along the poles. Steps I and II give the "mid-section“ of the graph. III. (HA) Use the last theorem to determine the behavior of the graph to the right and left as x±∞. The horizontal asymptote exists only if the limit exists. Step III gives the “arms” of the graph.
- 112. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1
- 113. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
- 114. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1.
- 115. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. x=2
- 116. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. x=2
- 117. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. + – + x=2 +
- 118. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. + – + + + x=2
- 119. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. + – + + + x=2
- 120. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. + – + + + x=2
- 121. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. + – + + + x=2
- 122. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. + – + + + x=2
- 123. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. As x±∞, R(x) resembles x2/x2 = 1, i.e. it has y = 1 + + – + + x=2 as the HA.
- 124. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. As x±∞, R(x) resembles x2/x2 = 1, i.e. it has y = 1 + + – + + x=2 as the HA.
- 125. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. As x±∞, R(x) resembles x2/x2 = 1, i.e. it has y = 1 + + – + + x=2 as the HA.
- 126. Graphs of Rational Functions Example B. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 4x + 4 x2 – 1 For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2). For VA, set Q(x) = 0, i.e.. x2 – 1 = 0 x = ± 1. All of them have order 1, so the sign changes at each of these values. + – + + + x=2 Note the y– int is (0, – 4) (0,–4) As x±∞, R(x) resembles x2/x2 = 1, i.e. it has y = 1 as the HA. Note the graph stays above the HA to the far left and below to the far right.
- 127. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2
- 128. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1.
- 129. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2.
- 130. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. Do the sign-chart.
- 131. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. x=3 Do the sign-chart.
- 132. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. x=3 Do the sign-chart. – + – +
- 133. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. Do the sign-chart. Construct the middle part of the graph. – + – + x=3
- 134. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. Do the sign-chart. Construct the middle part of the graph. – + – + x=3
- 135. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. Do the sign-chart. Construct the middle part of the graph. – + – + x=3
- 136. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. Do the sign-chart. Construct the middle part of the graph. – + – + x=3
- 137. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. Do the sign-chart. Construct the middle part of the graph. – + – + x=3
- 138. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. As x ±∞, the graph of R(x) resembles the graph of the quotient of the leading terms x2/x = x, or y = x. – + – + x=3 Do the sign-chart. Construct the middle part of the graph.
- 139. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. As x ±∞, the graph of R(x) resembles the graph of the quotient of the leading terms x2/x = x, or y = x. Hence there is no HA. – + – + x=3 Do the sign-chart. Construct the middle part of the graph.
- 140. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. As x ±∞, the graph of R(x) resembles the graph of the quotient of the leading terms x2/x = x, or y = x. Hence there is no HA. – + – + x=3 Do the sign-chart. Construct the middle part of the graph.
- 141. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. As x ±∞, the graph of R(x) resembles the graph of the quotient of the leading terms x2/x = x, or y = x. Hence there is no HA. x=3 Do the sign-chart. Construct the middle part of the graph. – + – +
- 142. Graphs of Rational Functions Example C. Find the roots, VA and HA, if any, of R(x) = Draw the sign-chart and sketch the graph. x2 – 2x – 3 x – 2 Set x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 so x = -1, 3 are the roots of order 1. For VA, set x – 2 = 0, i.e.. x = 2. As x ±∞, the graph of R(x) resembles the graph of the quotient of the leading terms x2/x = x, or y = x. Hence there is no HA. x=3 Do the sign-chart. Construct the middle part of the graph. (0, 3/2) – + – + Note the y– int (0, 3/2).

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