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11 x1 t10 01 graphing quadratics (2013)

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  • 1. The Quadratic Polynomialand the Parabola
  • 2. The Quadratic Polynomialand the ParabolaQuadratic polynomial –
  • 3. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c 
  • 4. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function –
  • 5. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  
  • 6. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation –
  • 7. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  
  • 8. The Quadratic Polynomialand the ParabolaQuadratic polynomial – 2ax bx c Quadratic function – 2y ax bx c  Quadratic equation – 20ax bx c  Coefficients –
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Graphing Quadratics
  • 21. Graphing QuadraticsThe graph of a quadratic function is a parabola.
  • 22. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  
  • 23. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  a
  • 24. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx
  • 25. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a 
  • 26. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up
  • 27. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave upyx
  • 28. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yx
  • 29. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave down
  • 30. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc
  • 31. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y intercept
  • 32. Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c  ayx0a concave up0a yxconcave downc = y interceptzeroes (roots)
  • 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. 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. 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. 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. 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. 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. 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. 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. e.g. 2Graph 8 12y x x  
  • 42. e.g. 2Graph 8 12y x x  a = 1 > 0yx
  • 43. e.g. 2Graph 8 12y x x  a = 1 > 0 concave upyx
  • 44. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12yx28 12y x x  
  • 45. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yyx
  • 46. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yyx12
  • 47. e.g. 2Graph 8 12y x x  a = 1 > 0 concave up c = 12  intercept is 0,12yzeroes yx12
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. (ii) Find the quadratic with;a) roots 3 and 6
  • 66. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x  
  • 67. (ii) Find the quadratic with;a) roots 3 and 6 29 18y a x x   6 3  6 3
  • 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. (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. (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. (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. (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. (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. (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. (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. (iii) Solve;2) 5 6 0a x x  
  • 77. (iii) Solve;2) 5 6 0a x x    2 3 0x x  
  • 78. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2
  • 79. (iii) Solve;2) 5 6 0a x x    2 3 0x x  yx–3 –2
  • 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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. Exercise 8A; 1adf, 2adf, 3bd, 4bd, 5c, 6ade, 7d, 9ace, 12c,13b, 14a