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
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)
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
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
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?