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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
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
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?