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 – 2
ax bx c 

4. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function –

5. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  

6. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation –

7. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  

8. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
Coefficients –

9. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –

10. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
Indeterminate –

11. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
xIndeterminate –

12. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
xIndeterminate –
Roots –

13. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
xIndeterminate –
Roots – Solutions to the quadratic equation

14. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
xIndeterminate –
Roots – Solutions to the quadratic equation
Zeroes –

15. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
xIndeterminate –
Roots – Solutions to the quadratic equation
Zeroes – x intercepts of the quadratic function

16. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
xIndeterminate –
Roots – Solutions to the quadratic equation
Zeroes – x intercepts of the quadratic function
2
e.g. Find the roots of 1 0x  

17. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
xIndeterminate –
Roots – Solutions to the quadratic equation
Zeroes – x intercepts of the quadratic function
2
e.g. Find the roots of 1 0x  
2
2
1 0
1
x
x
 


18. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
xIndeterminate –
Roots – Solutions to the quadratic equation
Zeroes – x intercepts of the quadratic function
2
e.g. Find the roots of 1 0x  
2
2
1 0
1
x
x
 

1x  

19. The Quadratic Polynomial
and the Parabola
Quadratic polynomial – 2
ax bx c 
Quadratic function – 2
y ax bx c  
Quadratic equation – 2
0ax bx c  
, ,a b cCoefficients –
xIndeterminate –
Roots – Solutions to the quadratic equation
Zeroes – x intercepts of the quadratic function
2
e.g. Find the roots of 1 0x  
2
2
1 0
1
x
x
 

1x   the roots are 1 and 1x x   

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. 2
y ax bx c  

23. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a

24. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x

25. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 

26. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up

27. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
y
x

28. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x

29. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down

30. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c

31. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept

32. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept
zeroes (roots)

33. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept
zeroes (roots) = x intercepts

34. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept
zeroes (roots) = x intercepts
2
b
x
a



35. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept
zeroes (roots) = x intercepts
2
b
x
a

 = axis of symmetry

36. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept
zeroes (roots) = x intercepts
2
b
x
a

 = axis of symmetry Note: AOS is the average of the zeroes

37. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept
zeroes (roots) = x intercepts
2
b
x
a

 = axis of symmetry Note: AOS is the average of the zeroes
vertex

38. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept
zeroes (roots) = x intercepts
2
b
x
a

 = axis of symmetry Note: AOS is the average of the zeroes
vertex x value is the AOS

39. Graphing Quadratics
The graph of a quadratic function is a parabola. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept
zeroes (roots) = x intercepts
2
b
x
a

 = axis of symmetry Note: AOS is the average of the zeroes
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. 2
y ax bx c  
a
y
x
0a 
concave up
0a 
y
x
concave down
c = y intercept
zeroes (roots) = x intercepts
2
b
x
a

 = axis of symmetry Note: AOS is the average of the zeroes
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. 2
Graph 8 12y x x  

42. e.g. 2
Graph 8 12y x x  
a = 1 > 0
y
x

43. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up
y
x

44. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12
y
x
2
8 12y x x  

45. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
y
x

46. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
y
x
12

47. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes y
x
12

48. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x   y
x
12
2
8 12y x x  

49. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
y
x
12
2
8 12y x x  

50. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
y
x
12
2
8 12y x x  

51. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
y
x
12
2
8 12y x x  

52. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
y
x
12
–2–6
2
8 12y x x  

53. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
y
x
12
–2–6
2
8 12y x x  

54. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


y
x
12
–2–6
2
8 12y x x  

55. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
y
x
12
–2–6
2
8 12y x x  

56. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
OR
6 2
2
x
 

y
x
12
–2–6
2
8 12y x x  

57. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
OR
6 2
2
x
 

4 
y
x
12
–2–6
2
8 12y x x  

58. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
OR
6 2
2
x
 

4 
y
x
12
–2–6
2
8 12y x x  

59. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
OR
6 2
2
x
 

4 
vertex
y
x
12
–2–6
2
8 12y x x  

60. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
OR
6 2
2
x
 

4 
vertex    
2
4 8 4 12y     
y
x
12
–2–6
2
8 12y x x  

61. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
OR
6 2
2
x
 

4 
vertex    
2
4 8 4 12y     
4 
y
x
12
–2–6
2
8 12y x x  

62. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
OR
6 2
2
x
 

4 
vertex    
2
4 8 4 12y     
4 
 vertex is 4, 4  
y
x
12
–2–6
2
8 12y x x  

63. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
OR
6 2
2
x
 

4 
vertex    
2
4 8 4 12y     
4 
 vertex is 4, 4  
y
x
12
–2–6
(–4, –4)
2
8 12y x x  

64. e.g. 2
Graph 8 12y x x  
a = 1 > 0 concave up c = 12  intercept is 0,12y
zeroes 2
8 12 0x x  
  6 2 0x x  
6 or 2x x   
   
intercepts are
6,0 and 2,0
x
 
AOS
2
b
x
a


8
2
4


 
OR
6 2
2
x
 

4 
vertex    
2
4 8 4 12y     
4 
 vertex is 4, 4  
y
x
12
–2–6
(–4, –4)
2
8 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
 2
9 18y a x x  

67. (ii) Find the quadratic with;
a) roots 3 and 6
 2
9 18y a x x  
 6 3  6 3

68. (ii) Find the quadratic with;
a) roots 3 and 6
 2
9 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
 2
9 18y a x x  
 6 3  6 3
b) monic roots 3 2 and 3 2 
2
6 7y x x  
c) roots 2 and 8 and vertex (5,3)
 2
10 16y a x x  
    2
5,3 : 3 5 10 5 16a  
3 9
1
3
a
a
 
 
 21
10 16
3
y x x    

70. (ii) Find the quadratic with;
a) roots 3 and 6
 2
9 18y a x x  
 6 3  6 3
b) monic roots 3 2 and 3 2 
2
6 7y x x  
 3 2 3 2   
  3 2 3 2 

71. (ii) Find the quadratic with;
a) roots 3 and 6
 2
9 18y a x x  
 6 3  6 3
b) monic roots 3 2 and 3 2 
2
6 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
 2
9 18y a x x  
 6 3  6 3
b) monic roots 3 2 and 3 2 
2
6 7y x x  
 3 2 3 2   
  3 2 3 2 
c) roots 2 and 8 and vertex (5,3)
 2
10 16y a x x  

73. (ii) Find the quadratic with;
a) roots 3 and 6
 2
9 18y a x x  
 6 3  6 3
b) monic roots 3 2 and 3 2 
2
6 7y x x  
 3 2 3 2   
  3 2 3 2 
c) roots 2 and 8 and vertex (5,3)
 2
10 16y a x x  
    2
5,3 : 3 5 10 5 16a  

74. (ii) Find the quadratic with;
a) roots 3 and 6
 2
9 18y a x x  
 6 3  6 3
b) monic roots 3 2 and 3 2 
2
6 7y x x  
 3 2 3 2   
  3 2 3 2 
c) roots 2 and 8 and vertex (5,3)
 2
10 16y a x x  
    2
5,3 : 3 5 10 5 16a  
3 9
1
3
a
a
 
 

75. (ii) Find the quadratic with;
a) roots 3 and 6
 2
9 18y a x x  
 6 3  6 3
b) monic roots 3 2 and 3 2 
2
6 7y x x  
 3 2 3 2   
  3 2 3 2 
c) roots 2 and 8 and vertex (5,3)
 2
10 16y a x x  
    2
5,3 : 3 5 10 5 16a  
3 9
1
3
a
a
 
 
 21
10 16
3
y 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  
y
x–3 –2

79. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2

80. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?

81. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?

82. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?
3 or 2x x   

83. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola 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  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?
3 or 2x x   
2
) 3 4b x x   
2
3 4 0x x  

85. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?
3 or 2x x   
2
) 3 4b x x   
  4 1 0x x  
2
3 4 0x x  

86. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?
3 or 2x x   
2
) 3 4b x x   
  4 1 0x x  
y
x1–4
2
3 4 0x x  

87. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?
3 or 2x x   
2
) 3 4b x x   
  4 1 0x x  
y
x1–4
2
3 4 0x x  

88. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?
3 or 2x x   
2
) 3 4b x x   
  4 1 0x x  
y
x1–4
Q: for what values of x is the
parabola below the x axis?
2
3 4 0x x  

89. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?
3 or 2x x   
2
) 3 4b x x   
  4 1 0x x  
y
x1–4
Q: for what values of x is the
parabola below the x axis?
2
3 4 0x x  

90. (iii) Solve;
2
) 5 6 0a x x  
  2 3 0x x  
y
x–3 –2
Q: for what values of x is the
parabola above the x axis?
3 or 2x x   
2
) 3 4b x x   
  4 1 0x x  
y
x1–4
Q: for what values of x is the
parabola below the x axis?
4 1x  
2
3 4 0x x  

91. Exercise 8A; 1adf, 2adf, 3bd, 4bd, 5c, 6ade, 7d, 9ace, 12c,
13b, 14a