11 x1 t10 01 graphing quadratics (2013)

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11 x1 t10 01 graphing quadratics (2013)

  1. 1. The Quadratic Polynomialand the Parabola
  2. 2. The Quadratic Polynomialand the ParabolaQuadratic polynomial –
  3. 3. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c 
  4. 4. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function –
  5. 5. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  
  6. 6. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation –
  7. 7. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  
  8. 8. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  Coefficients –
  9. 9. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –
  10. 10. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –Indeterminate –
  11. 11. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –xIndeterminate –
  12. 12. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –xIndeterminate –Roots –
  13. 13. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –xIndeterminate –Roots – Solutions to the quadratic equation
  14. 14. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –xIndeterminate –Roots – Solutions to the quadratic equationZeroes –
  15. 15. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –xIndeterminate –Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function
  16. 16. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –xIndeterminate –Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function2e.g. Find the roots of 1 0x  
  17. 17. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –xIndeterminate –Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function2e.g. Find the roots of 1 0x  221 01xx 
  18. 18. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –xIndeterminate –Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function2e.g. Find the roots of 1 0x  221 01xx 1x  
  19. 19. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  , ,a b cCoefficients –xIndeterminate –Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function2e.g. Find the roots of 1 0x  221 01xx 1x   the roots are 1 and 1x x   
  20. 20. Graphing Quadratics
  21. 21. Graphing QuadraticsThe graph of a quadratic function is a parabola.
  22. 22. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  
  23. 23. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  a
  24. 24. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx
  25. 25. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a 
  26. 26. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up
  27. 27. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave upyx
  28. 28. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yx
  29. 29. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave down
  30. 30. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc
  31. 31. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y intercept
  32. 32. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots)
  33. 33. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots) = x intercepts
  34. 34. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots) = x intercepts2bxa
  35. 35. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots) = x intercepts2bxa = axis of symmetry
  36. 36. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots) = x intercepts2bxa = axis of symmetry Note: AOS is the average of the zeroes
  37. 37. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots) = x intercepts2bxa = axis of symmetry Note: AOS is the average of the zeroesvertex
  38. 38. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots) = x intercepts2bxa = axis of symmetry Note: AOS is the average of the zeroesvertex x value is the AOS
  39. 39. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots) = x intercepts2bxa = axis of symmetry Note: AOS is the average of the zeroesvertex x value is the AOSy value is found by substituting AOS into the function.
  40. 40. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots) = x intercepts2bxa = axis of symmetry Note: AOS is the average of the zeroesvertex x value is the AOSy value is found by substituting AOS into the function.(It is the maximum/minimum value of the function)
  41. 41. e.g. 2Graph 8 12y x x  
  42. 42. e.g. 2Graph 8 12y x x  a = 1 > 0yx
  43. 43. e.g. 2Graph 8 12y x x  a = 1 > 0 concave upyx
  44. 44. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12yx28 12y x x  
  45. 45. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yyx
  46. 46. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yyx12
  47. 47. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes yx12
  48. 48. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x   yx1228 12y x x  
  49. 49. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  yx1228 12y x x  
  50. 50. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x   yx1228 12y x x  
  51. 51. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x yx1228 12y x x  
  52. 52. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x yx12–2–628 12y x x  
  53. 53. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOSyx12–2–628 12y x x  
  54. 54. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxayx12–2–628 12y x x  
  55. 55. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 yx12–2–628 12y x x  
  56. 56. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 OR6 22x yx12–2–628 12y x x  
  57. 57. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 OR6 22x 4 yx12–2–628 12y x x  
  58. 58. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 OR6 22x 4 yx12–2–628 12y x x  
  59. 59. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 OR6 22x 4 vertexyx12–2–628 12y x x  
  60. 60. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 OR6 22x 4 vertex    24 8 4 12y     yx12–2–628 12y x x  
  61. 61. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 OR6 22x 4 vertex    24 8 4 12y     4 yx12–2–628 12y x x  
  62. 62. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 OR6 22x 4 vertex    24 8 4 12y     4  vertex is 4, 4  yx12–2–628 12y x x  
  63. 63. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 OR6 22x 4 vertex    24 8 4 12y     4  vertex is 4, 4  yx12–2–6(–4, –4)28 12y x x  
  64. 64. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes 28 12 0x x    6 2 0x x  6 or 2x x      intercepts are6,0 and 2,0x AOS2bxa824 OR6 22x 4 vertex    24 8 4 12y     4  vertex is 4, 4  yx12–2–6(–4, –4)28 12y x x  
  65. 65. (ii) Find the quadratic with;a) roots 3 and 6
  66. 66. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x  
  67. 67. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x   6 3  6 3
  68. 68. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x   6 3  6 3b) monic roots 3 2 and 3 2 
  69. 69. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x   6 3  6 3b) monic roots 3 2 and 3 2 26 7y x x  c) roots 2 and 8 and vertex (5,3) 210 16y a x x      25,3 : 3 5 10 5 16a  3 913aa   2110 163y x x    
  70. 70. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x   6 3  6 3b) monic roots 3 2 and 3 2 26 7y x x   3 2 3 2     3 2 3 2 
  71. 71. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x   6 3  6 3b) monic roots 3 2 and 3 2 26 7y x x   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 29 18y a x x   6 3  6 3b) monic roots 3 2 and 3 2 26 7y x x   3 2 3 2     3 2 3 2 c) roots 2 and 8 and vertex (5,3) 210 16y a x x  
  73. 73. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x   6 3  6 3b) monic roots 3 2 and 3 2 26 7y x x   3 2 3 2     3 2 3 2 c) roots 2 and 8 and vertex (5,3) 210 16y a x x      25,3 : 3 5 10 5 16a  
  74. 74. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x   6 3  6 3b) monic roots 3 2 and 3 2 26 7y x x   3 2 3 2     3 2 3 2 c) roots 2 and 8 and vertex (5,3) 210 16y a x x      25,3 : 3 5 10 5 16a  3 913aa  
  75. 75. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x   6 3  6 3b) monic roots 3 2 and 3 2 26 7y x x   3 2 3 2     3 2 3 2 c) roots 2 and 8 and vertex (5,3) 210 16y a x x      25,3 : 3 5 10 5 16a  3 913aa   2110 163y x x    
  76. 76. (iii) Solve;2) 5 6 0a x x  
  77. 77. (iii) Solve;2) 5 6 0a x x    2 3 0x x  
  78. 78. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2
  79. 79. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2
  80. 80. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?
  81. 81. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?
  82. 82. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?3 or 2x x   
  83. 83. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?3 or 2x x   2) 3 4b x x   
  84. 84. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?3 or 2x x   2) 3 4b x x   23 4 0x x  
  85. 85. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?3 or 2x x   2) 3 4b x x     4 1 0x x  23 4 0x x  
  86. 86. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?3 or 2x x   2) 3 4b x x     4 1 0x x  yx1–423 4 0x x  
  87. 87. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?3 or 2x x   2) 3 4b x x     4 1 0x x  yx1–423 4 0x x  
  88. 88. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?3 or 2x x   2) 3 4b x x     4 1 0x x  yx1–4Q: for what values of x is theparabola below the x axis?23 4 0x x  
  89. 89. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?3 or 2x x   2) 3 4b x x     4 1 0x x  yx1–4Q: for what values of x is theparabola below the x axis?23 4 0x x  
  90. 90. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2Q: for what values of x is theparabola above the x axis?3 or 2x x   2) 3 4b x x     4 1 0x x  yx1–4Q: for what values of x is theparabola below the x axis?4 1x  23 4 0x x  
  91. 91. Exercise 8A; 1adf, 2adf, 3bd, 4bd, 5c, 6ade, 7d, 9ace, 12c,13b, 14a

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