The Quadratic Polynomial
and the Parabola

The Quadratic Polynomial
and the Parabola
Quadratic polynomial –

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

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

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

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

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

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 –

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

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 –

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

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 –

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

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 –

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

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

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

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 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

Graphing Quadratics

Graphing Quadratics
The graph of a quadratic function is a parabola.

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

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

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

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

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

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

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

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

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

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

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)

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

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

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

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

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

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

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.

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)

e.g. Graph y  x 2  8 x  12

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 

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 

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 

(ii) Find the quadratic with;
a) roots 3 and 6

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

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

(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

(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

(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 

(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)

(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 

(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 

(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

(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

(iii) Solve;
a) x 2  5 x  6  0

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

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

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

(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?

(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?

(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?

(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

(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

(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

(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

(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

(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?

(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?

(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?

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