2.6 graphs of factorable polynomials

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2.6 graphs of factorable polynomials

  1. 1. Graphs of Factorable Polynomials http://www.lahc.edu/math/precalculus/math_260a.html
  2. 2. Graphs of Factorable Polynomials In this and the next sections, we devise a strategy for graphing factorable polynomial and rational functions.
  3. 3. Graphs of Factorable Polynomials In this and the next sections, we devise a strategy for graphing factorable polynomial and rational functions. A basic fact about the graphs of polynomials is that their graphs are continuous lines: x x Graphs of Polynomials
  4. 4. Graphs of Factorable Polynomials In this and the next sections, we devise a strategy for graphing factorable polynomial and rational functions. A basic fact about the graphs of polynomials is that their graphs are continuous lines: x x Graphs of Polynomials x x Graphs of Non–polynomials
  5. 5. Graphs of Factorable Polynomials In this and the next sections, we devise a strategy for graphing factorable polynomial and rational functions. A basic fact about the graphs of polynomials is that their graphs are continuous lines: x x Graphs of Polynomials x x Graphs of Non–polynomials To graph a function, we separate the job into two parts.
  6. 6. Graphs of Factorable Polynomials In this and the next sections, we devise a strategy for graphing factorable polynomial and rational functions. A basic fact about the graphs of polynomials is that their graphs are continuous lines: x x Graphs of Polynomials x x Graphs of Non–polynomials To graph a function, we separate the job into two parts. I. What does the graph look like in the “middle”, i.e. the curvy portion that we draw on papers?
  7. 7. Graphs of Factorable Polynomials In this and the next sections, we devise a strategy for graphing factorable polynomial and rational functions. A basic fact about the graphs of polynomials is that their graphs are continuous lines: x x Graphs of Polynomials x x Graphs of Non–polynomials To graph a function, we separate the job into two parts. I. What does the graph look like in the “middle”, i.e. the curvy portion that we draw on papers? II. What does the graph look like beyond the drawn area?
  8. 8. Graphs of Factorable Polynomials In this and the next sections, we devise a strategy for graphing factorable polynomial and rational functions. A basic fact about the graphs of polynomials is that their graphs are continuous lines: x x Graphs of Polynomials x x Graphs of Non–polynomials To graph a function, we separate the job into two parts. I. What does the graph look like in the “middle”, i.e. the curvy portion that we draw on papers? II. What does the graph look like beyond the drawn area? We start with part II with factorable polynomials.
  9. 9. Graphs of Factorable Polynomials We start with the graphs of the polynomials y = ±xN.
  10. 10. Graphs of Factorable Polynomials We start with the graphs of the polynomials y = ±xN. The graphs y = xeven y = x2
  11. 11. Graphs of Factorable Polynomials The graphs y = xeven y = x2y = x4 (1, 1)(-1, 1) We start with the graphs of the polynomials y = ±xN.
  12. 12. Graphs of Factorable Polynomials The graphs y = xeven y = x2y = x4y = x6 (1, 1)(-1, 1) We start with the graphs of the polynomials y = ±xN.
  13. 13. Graphs of Factorable Polynomials The graphs y = xeven y = x2y = x4y = x6 y = -x2 y = -x4 y = -x6 (1, 1)(-1, 1) (-1,-1) (1,-1) We start with the graphs of the polynomials y = ±xN.
  14. 14. Graphs of Factorable Polynomials The graphs y = xeven y = x2y = x4y = x6 y = -x2 y = -x4 y = -x6 y = ±xeven: y = xeven y = –xeven (1, 1)(-1, 1) (-1,-1) (1,-1) We start with the graphs of the polynomials y = ±xN.
  15. 15. Graphs of Factorable Polynomials y = x3 The graphs y = xodd
  16. 16. Graphs of Factorable Polynomials y = x3 y = x5 (1, 1) (-1, -1) The graphs y = xodd
  17. 17. Graphs of Factorable Polynomials y = x3 y = x5 y = x7 (1, 1) (-1, -1) The graphs y = xodd
  18. 18. Graphs of Factorable Polynomials The graphs y = xodd y = x3 y = x5 y = x7 y = -x3 y = -x5 y = -x7 (1, 1) (-1, -1) (-1, 1) (1,-1)
  19. 19. Graphs of Factorable Polynomials The graphs y = xodd y = x3 y = x5 y = x7 y = -x3 y = -x5 y = -x7 y = ±xodd y = xodd y = –xodd (1, 1) (-1, -1) (-1, 1) (1,-1)
  20. 20. Graphs of Factorable Polynomials Facts about the graphs of polynomials:
  21. 21. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves.
  22. 22. Graphs of Factorable Polynomials Facts about the graphs of polynomials: • The graphs of polynomials are unbroken curves. • Polynomial curves are smooth (no corners).
  23. 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.
  24. 24. 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.
  25. 25. 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" is negligible compare to anxn.
  26. 26. 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" is negligible compare to anxn. Hence, for x where |x| is "large", the graph of P(x) resembles the graph y = anxn.
  27. 27. 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" is negligible compare 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).
  28. 28. 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" is negligible compare 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). These behaviors are based on the sign the leading term anxn, and whether n is even or odd.
  29. 29. Graphs of Factorable Polynomials This behaviors of polynomial-graphs to the "sides":
  30. 30. Graphs of Factorable Polynomials y = +xeven + lower degree terms: This behaviors of polynomial-graphs to the "sides":
  31. 31. Graphs of Factorable Polynomials y = +xeven + lower degree terms: y = –xeven + lower degree terms: This behaviors of polynomial-graphs to the "sides":
  32. 32. Graphs of Factorable Polynomials y = +xeven + lower degree terms: y = –xeven + lower degree terms: This behaviors of polynomial-graphs to the "sides": y = +xodd + lower degree terms:
  33. 33. Graphs of Factorable Polynomials y = +xeven + lower degree terms: y = –xeven + lower degree terms: This behaviors of polynomial-graphs to the "sides": y = +xodd + lower degree terms: y = –xodd + lower degree terms:
  34. 34. Graphs of Factorable Polynomials For factorable polynomials, we use the sign-charts to sketch the central portion of the graphs.
  35. 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:
  36. 36. 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):
  37. 37. 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.
  38. 38. 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.
  39. 39. 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 it's sign, use the orders of the roots to extend and fill in the signs.
  40. 40. 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 it's sign, use 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.
  41. 41. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Graphs of Factorable Polynomials
  42. 42. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 Graphs of Factorable Polynomials
  43. 43. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Graphs of Factorable Polynomials
  44. 44. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. Graphs of Factorable Polynomials
  45. 45. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. 4-1 Graphs of Factorable Polynomials
  46. 46. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. Test for sign using x = 0 and we get f(0) negative. 4-1 Graphs of Factorable Polynomials
  47. 47. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. Test for sign using x = 0 and we get f(0) negative. 0 4-1 Test x = 0, we get that f(0) negative. Graphs of Factorable Polynomials
  48. 48. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. Test for sign using x = 0 and we get f(0) negative. Since both roots are odd-ordered, the sign changes to "+" across the them. 0 4-1 Test x = 0, we get that f(0) negative. Graphs of Factorable Polynomials
  49. 49. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. Test for sign using x = 0 and we get f(0) negative. Since both roots are odd-ordered, the sign changes to "+" across the them. 0 4-1 Test x = 0, we get that f(0) negative. Graphs of Factorable Polynomials + +
  50. 50. Example A. Make the sign-chart of f(x) = x2 – 3x – 4 Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0 The roots are x = 4 , -1 and both are odd-ordered. Mark off these points on a line. Test for sign using x = 0 and we get f(0) negative. Since both roots are odd-ordered, the sign changes to "+" across the them. 0 4-1 Test x = 0, we get that f(0) negative. Graphs of Factorable Polynomials The graph of y = x2 – 3x – 4 is shown below: + +
  51. 51. Graphs of Factorable Polynomials + + + + + – – – – – + + + + + 4-1 y=(x – 4)(x+1)
  52. 52. Graphs of Factorable Polynomials Note the sign-chart reflects the graph: I. The graph touches or crosses the x-axis at the roots. + + + + + – – – – – + + + + + 4-1 y=(x – 4)(x+1)
  53. 53. Graphs of Factorable Polynomials Note the sign-chart reflects 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 "+". + + + + + – – – – – + + + + + 4-1 y=(x – 4)(x+1)
  54. 54. Graphs of Factorable Polynomials Note the sign-chart reflects 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 "–". + + + + + – – – – – + + + + + 4-1 y=(x – 4)(x+1)
  55. 55. Graphs of Factorable Polynomials II. The “Mid-Portions” of Polynomial Graphs
  56. 56. Graphs of Factorable Polynomials II. The “Mid-Portions” of Polynomial Graphs Graphs of an odd ordered root (x – r)odd at x = r.
  57. 57. Graphs of Factorable Polynomials + + order = 1 order = 1 r r II. The “Mid-Portions” of Polynomial Graphs Graphs of an odd ordered root (x – r)odd at x = r. y = (x – r)1 y = –(x – r)1
  58. 58. Graphs of Factorable Polynomials + + + order = 1 order = 1 order = 3, 5, 7.. r r r II. The “Mid-Portions” of Polynomial Graphs Graphs of an odd ordered root (x – r)odd at x = r. + order = 3, 5, 7.. r y = (x – r)1 y = –(x – r)1 y = (x – r)3 or 5.. y = –(x – r)3 or 5..
  59. 59. Graphs of Factorable Polynomials order = 2, 4, 6 .. ++ x=r r Graphs of an even ordered root at (x – r)even at x = r. order = 2, 4, 6 .. y = (x – r)2 or 4.. y = –(x – r)2 or 4..
  60. 60. Graphs of Factorable Polynomials order = 2, 4, 6 .. ++ 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. x=r r Graphs of an even ordered root at (x – r)even at x = r. order = 2, 4, 6 .. y = (x – r)2 or 4.. y = –(x – r)2 or 4..
  61. 61. Graphs of Factorable Polynomials order = 2, 4, 6 .. ++ 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. x=r r order = 2, 4, 6 .. y = (x – r)2 or 4.. y = –(x – r)2 or 4.. Graphs of an even ordered root at (x – r)even at x = r.
  62. 62. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial,
  63. 63. Graphs of Factorable Polynomials For example, given two roots with their orders and the sign-chart of a polynomial, + order=2 order=3
  64. 64. 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
  65. 65. 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
  66. 66. 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
  67. 67. 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
  68. 68. 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
  69. 69. 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
  70. 70. 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
  71. 71. 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
  72. 72. 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.
  73. 73. 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.
  74. 74. 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.)
  75. 75. 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).
  76. 76. 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).
  77. 77. Graphs of Factorable Polynomials Note the graph resembles its leading term: y = –x5, when viewed at a distance:
  78. 78. Graphs of Factorable Polynomials Note the graph resembles its leading term: y = –x5, when viewed at a distance: -2 ++ 0 3

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