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11X1 T11 01 graphing quadratics

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11X1 T11 01 graphing quadratics

  1. 1. The Quadratic Polynomial and the Parabola
  2. 2. The Quadratic Polynomial and the Parabola Quadratic polynomial –
  3. 3. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c
  4. 4. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function –
  5. 5. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c
  6. 6. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation –
  7. 7. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0
  8. 8. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients –
  9. 9. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c
  10. 10. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate –
  11. 11. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate – x
  12. 12. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate – x Roots –
  13. 13. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate – x Roots – Solutions to the quadratic equation
  14. 14. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate – x Roots – Solutions to the quadratic equation Zeroes –
  15. 15. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate – x Roots – Solutions to the quadratic equation Zeroes – x intercepts of the quadratic function
  16. 16. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate – x Roots – Solutions to the quadratic equation Zeroes – x intercepts of the quadratic function e.g. Find the roots of x 2  1  0
  17. 17. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate – x Roots – Solutions to the quadratic equation Zeroes – x intercepts of the quadratic function e.g. Find the roots of x 2  1  0 x2 1  0 x2  1
  18. 18. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate – x Roots – Solutions to the quadratic equation Zeroes – x intercepts of the quadratic function e.g. Find the roots of x 2  1  0 x2 1  0 x2  1 x  1
  19. 19. The Quadratic Polynomial and the Parabola Quadratic polynomial – ax 2  bx  c Quadratic function – y  ax 2  bx  c Quadratic equation – ax 2  bx  c  0 Coefficients – a, b, c Indeterminate – x Roots – Solutions to the quadratic equation Zeroes – x intercepts of the quadratic function e.g. Find the roots of x 2  1  0 x2 1  0 x2  1 x  1  the roots are x  1 and x  1
  20. 20. Graphing Quadratics
  21. 21. Graphing Quadratics The graph of a quadratic function is a parabola.
  22. 22. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c
  23. 23. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c a
  24. 24. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y a x
  25. 25. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y a x a0
  26. 26. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y a x a0 concave up
  27. 27. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 concave up
  28. 28. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up
  29. 29. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down
  30. 30. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c
  31. 31. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept
  32. 32. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots)
  33. 33. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts
  34. 34. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts b x 2a
  35. 35. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts b x = axis of symmetry 2a
  36. 36. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts b x = axis of symmetry Note: AOS is the average of the zeroes 2a
  37. 37. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts b x = axis of symmetry Note: AOS is the average of the zeroes 2a vertex
  38. 38. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts b x = axis of symmetry Note: AOS is the average of the zeroes 2a vertex x value is the AOS
  39. 39. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts b x = axis of symmetry Note: AOS is the average of the zeroes 2a vertex x value is the AOS y value is found by substituting AOS into the function.
  40. 40. Graphing Quadratics The graph of a quadratic function is a parabola. y  ax 2  bx  c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts b x = axis of symmetry Note: AOS is the average of the zeroes 2a vertex x value is the AOS y value is found by substituting AOS into the function. (It is the maximum/minimum value of the function)
  41. 41. e.g. Graph y  x 2  8 x  12
  42. 42. e.g. Graph y  x 2  8 x  12 a=1>0 y x
  43. 43. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up y x
  44. 44. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12 y y  x 2  8 x  12 x
  45. 45. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  y x
  46. 46. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  y 12 x
  47. 47. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes y 12 x
  48. 48. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12 12 x
  49. 49. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x
  50. 50. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2 x
  51. 51. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  x
  52. 52. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  –6 –2 x
  53. 53. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS –6 –2 x
  54. 54. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b 2a –6 –2 x
  55. 55. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b 2a 8  2 –6 –2 x  4
  56. 56. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b OR x  6  2 2a 2 8  2 –6 –2 x  4
  57. 57. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b OR x  6  2 2a 2 8  4  2 –6 –2 x  4
  58. 58. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b OR x  6  2 2a 2 8  4  2 –6 –2 x  4
  59. 59. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b OR x  6  2 2a 2 8  4  2 –6 –2 x  4 vertex
  60. 60. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b OR x  6  2 2a 2 8  4  2 –6 –2 x  4 y   4   8  4   12 2 vertex
  61. 61. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b OR x  6  2 2a 2 8  4  2 –6 –2 x  4 y   4   8  4   12 2 vertex  4
  62. 62. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b OR x  6  2 2a 2 8  4  2 –6 –2 x  4 y   4   8  4   12 2 vertex  4  vertex is  4, 4 
  63. 63. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b OR x  6  2 2a 2 8  4  2 –6 –2 x  4 y   4   8  4   12 2 vertex (–4, –4)  4  vertex is  4, 4 
  64. 64. e.g. Graph y  x 2  8 x  12 a = 1 > 0  concave up c = 12  y intercept is  0,12  zeroes x 2  8 x  12  0 y y  x 2  8 x  12  x  6  x  2   0 12 x  6 or x  2  x intercepts are  6,0  and  2,0  AOS x  b OR x  6  2 2a 2 8  4  2 –6 –2 x  4 y   4   8  4   12 2 vertex (–4, –4)  4  vertex is  4, 4 
  65. 65. (ii) Find the quadratic with; a) roots 3 and 6
  66. 66. (ii) Find the quadratic with; a) roots 3 and 6 y  a  x 2  9 x  18 
  67. 67. (ii) Find the quadratic with; a) roots 3 and 6 y  a  x 2  9 x  18    6  3 63
  68. 68. (ii) Find the quadratic with; a) roots 3 and 6 b) monic roots 3  2 and 3  2 y  a  x 2  9 x  18    6  3 63
  69. 69. (ii) Find the quadratic with; a) roots 3 and 6 b) monic roots 3  2 and 3  2 y  a  x 2  9 x  18  y  x2  6x  7   6  3 63 c) roots 2 and 8 and vertex (5,3) y  a  x 2  10 x  16   5,3 : 3  a  52  10  5   16  3  9a 1 a 3  y    x  10 x  16  1 2 3
  70. 70. (ii) Find the quadratic with; a) roots 3 and 6 b) monic roots 3  2 and 3  2 y  a  x 2  9 x  18  y  x2  6x  7   6  3 63   3 2 3 2  3  2 3  2 
  71. 71. (ii) Find the quadratic with; a) roots 3 and 6 b) monic roots 3  2 and 3  2 y  a  x 2  9 x  18  y  x2  6x  7   6  3 63   3 2 3 2  3  2 3  2  c) roots 2 and 8 and vertex (5,3)
  72. 72. (ii) Find the quadratic with; a) roots 3 and 6 b) monic roots 3  2 and 3  2 y  a  x 2  9 x  18  y  x2  6x  7   6  3 63   3 2 3 2  3  2 3  2  c) roots 2 and 8 and vertex (5,3) y  a  x 2  10 x  16 
  73. 73. (ii) Find the quadratic with; a) roots 3 and 6 b) monic roots 3  2 and 3  2 y  a  x 2  9 x  18  y  x2  6x  7   6  3 63   3 2 3 2  3  2 3  2  c) roots 2 and 8 and vertex (5,3) y  a  x 2  10 x  16   5,3 : 3  a  52  10  5   16 
  74. 74. (ii) Find the quadratic with; a) roots 3 and 6 b) monic roots 3  2 and 3  2 y  a  x 2  9 x  18  y  x2  6x  7   6  3 63   3 2 3 2  3  2 3  2  c) roots 2 and 8 and vertex (5,3) y  a  x 2  10 x  16   5,3 : 3  a  52  10  5   16  3  9a 1 a 3
  75. 75. (ii) Find the quadratic with; a) roots 3 and 6 b) monic roots 3  2 and 3  2 y  a  x 2  9 x  18  y  x2  6x  7   6  3 63   3 2 3 2  3  2 3  2  c) roots 2 and 8 and vertex (5,3) y  a  x 2  10 x  16   5,3 : 3  a  52  10  5   16  3  9a 1 a 3  y    x  10 x  16  1 2 3
  76. 76. (iii) Solve; a) x 2  5 x  6  0
  77. 77. (iii) Solve; a) x 2  5 x  6  0  x  2  x  3  0
  78. 78. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 –3 –2 x
  79. 79. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 –3 –2 x
  80. 80. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 –3 –2 x Q: for what values of x is the parabola above the x axis?
  81. 81. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 –3 –2 x Q: for what values of x is the parabola above the x axis?
  82. 82. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 x  3 or x  2 –3 –2 x Q: for what values of x is the parabola above the x axis?
  83. 83. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 x  3 or x  2 –3 –2 x Q: for what values of x is the parabola above the x axis? b)  x 2  3 x  4
  84. 84. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 x  3 or x  2 –3 –2 x Q: for what values of x is the parabola above the x axis? b)  x 2  3 x  4 x 2  3x  4  0
  85. 85. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 x  3 or x  2 –3 –2 x Q: for what values of x is the parabola above the x axis? b)  x 2  3 x  4 x 2  3x  4  0  x  4  x  1  0
  86. 86. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 x  3 or x  2 –3 –2 x Q: for what values of x is the parabola above the x axis? b)  x 2  3 x  4 y x 2  3x  4  0  x  4  x  1  0 –4 1 x
  87. 87. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 x  3 or x  2 –3 –2 x Q: for what values of x is the parabola above the x axis? b)  x 2  3 x  4 y x 2  3x  4  0  x  4  x  1  0 –4 1 x
  88. 88. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 x  3 or x  2 –3 –2 x Q: for what values of x is the parabola above the x axis? b)  x 2  3 x  4 y x 2  3x  4  0  x  4  x  1  0 –4 1 x Q: for what values of x is the parabola below the x axis?
  89. 89. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 x  3 or x  2 –3 –2 x Q: for what values of x is the parabola above the x axis? b)  x 2  3 x  4 y x 2  3x  4  0  x  4  x  1  0 –4 1 x Q: for what values of x is the parabola below the x axis?
  90. 90. (iii) Solve; a) x 2  5 x  6  0 y  x  2  x  3  0 x  3 or x  2 –3 –2 x Q: for what values of x is the parabola above the x axis? b)  x 2  3 x  4 y x 2  3x  4  0  x  4  x  1  0 4  x  1 –4 1 x Q: for what values of x is the parabola below the x axis?
  91. 91. Exercise 8A; 1adf, 2adf, 3bd, 4bd, 5c, 6ade, 7d, 9ace, 12c, 13b, 14a

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