11X1 T11 01 graphing quadratics

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  • 1. The Quadratic Polynomial and the Parabola
  • 2. The Quadratic Polynomial and the Parabola Quadratic polynomial
  • 3. The Quadratic Polynomial and the Parabola Quadratic polynomial ax 2 bx c
  • 4. The Quadratic Polynomial and the Parabola Quadratic polynomial ax 2 bx c Quadratic function
  • 5. The Quadratic Polynomial and the Parabola Quadratic polynomial ax 2 bx c Quadratic function y ax 2 bx c
  • 6. The Quadratic Polynomial and the Parabola Quadratic polynomial ax 2 bx c Quadratic function y ax 2 bx c Quadratic equation
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Graphing Quadratics
  • 21. Graphing Quadratics The graph of a quadratic function is a parabola.
  • 22. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c
  • 23. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c a
  • 24. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y a x
  • 25. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y a x a0
  • 26. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y a x a0 concave up
  • 27. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 concave up
  • 28. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 concave up
  • 29. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 concave up concave down
  • 30. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 concave up concave down c
  • 31. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 concave up concave down c = y intercept
  • 32. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots)
  • 33. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts
  • 34. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts b x 2a
  • 35. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 concave up concave down c = y intercept zeroes (roots) = x intercepts b x = axis of symmetry 2a
  • 36. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 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. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 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. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 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. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 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. Graphing Quadratics The graph of a quadratic function is a parabola. y ax 2 bx c y y a x x a0 a0 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. e.g. Graph y x 2 8 x 12
  • 42. e.g. Graph y x 2 8 x 12 a=1>0 y x
  • 43. e.g. Graph y x 2 8 x 12 a = 1 > 0 concave up y x
  • 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. e.g. Graph y x 2 8 x 12 a = 1 > 0 concave up c = 12 y intercept is 0,12 y x
  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. (ii) Find the quadratic with; a) roots 3 and 6
  • 66. (ii) Find the quadratic with; a) roots 3 and 6 y a x 2 9 x 18
  • 67. (ii) Find the quadratic with; a) roots 3 and 6 y a x 2 9 x 18 6 3 63
  • 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 63
  • 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 63 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. (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 63 3 2 3 2 3 2 3 2
  • 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 63 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 b) monic roots 3 2 and 3 2 y a x 2 9 x 18 y x2 6x 7 6 3 63 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. (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 63 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. (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 63 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. (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 63 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. (iii) Solve; a) x 2 5 x 6 0
  • 77. (iii) Solve; a) x 2 5 x 6 0 x 2 x 3 0
  • 78. (iii) Solve; a) x 2 5 x 6 0 y x 2 x 3 0 3 2 x
  • 79. (iii) Solve; a) x 2 5 x 6 0 y x 2 x 3 0 3 2 x
  • 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. (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. (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. (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. (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. (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. (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. (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. (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. (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. (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. Exercise 8A; 1adf, 2adf, 3bd, 4bd, 5c, 6ade, 7d, 9ace, 12c, 13b, 14a