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The Quadratic Polynomial and the Parabola
The Quadratic Polynomial and the Parabola
Quadratic polynomial –
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function –
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation –
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c Coefficients –
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –
Indeterminate –
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –xIndeterminate –
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –xIndeterminate –
Roots –
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –xIndeterminate –
Roots – Solutions to the quadratic equation
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –xIndeterminate –
Roots – Solutions to the quadratic equationZeroes –
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –xIndeterminate –
Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –xIndeterminate –
Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function
2e.g. Find the roots of 1 0x
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –xIndeterminate –
Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function
2e.g. Find the roots of 1 0x 2
2
1 01
xx
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –xIndeterminate –
Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function
2e.g. Find the roots of 1 0x 2
2
1 01
xx
1x
The Quadratic Polynomial and the Parabola
Quadratic polynomial – 2ax bx c
Quadratic function – 2y ax bx c
Quadratic equation – 2 0ax bx c , ,a b cCoefficients –xIndeterminate –
Roots – Solutions to the quadratic equationZeroes – x intercepts of the quadratic function
2e.g. Find the roots of 1 0x 2
2
1 01
xx
1x the roots are 1 and 1x x
Graphing Quadratics
Graphing QuadraticsThe graph of a quadratic function is a parabola.
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
a
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up
y
x
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave down
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y intercept
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y interceptzeroes (roots)
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y interceptzeroes (roots) = x intercepts
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y interceptzeroes (roots) = x intercepts
2bxa
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y interceptzeroes (roots) = x intercepts
2bxa
= axis of symmetry
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y interceptzeroes (roots) = x intercepts
2bxa
= axis of symmetry Note: AOS is the average of the zeroes
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y interceptzeroes (roots) = x intercepts
2bxa
= axis of symmetry Note: AOS is the average of the zeroes
vertex
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y interceptzeroes (roots) = x intercepts
2bxa
= axis of symmetry Note: AOS is the average of the zeroes
vertex x value is the AOS
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y interceptzeroes (roots) = x intercepts
2bxa
= axis of symmetry Note: AOS is the average of the zeroes
vertex x value is the AOSy value is found by substituting AOS into the function.
Graphing QuadraticsThe graph of a quadratic function is a parabola. 2y ax bx c
ay
x0a
concave up0a
y
x
concave downc = y interceptzeroes (roots) = x intercepts
2bxa
= axis of symmetry Note: AOS is the average of the zeroes
vertex x value is the AOSy value is found by substituting AOS into the function.
(It is the maximum/minimum value of the function)
e.g. 2Graph 8 12y x x
e.g. 2Graph 8 12y x x a = 1 > 0
y
x
e.g. 2Graph 8 12y x x a = 1 > 0 concave up
y
x
e.g. 2Graph 8 12y x x a = 1 > 0 concave up c = 12
y
x
2 8 12y x x
e.g. 2Graph 8 12y x x a = 1 > 0 concave up c = 12 intercept is 0,12y
y
x
e.g. 2Graph 8 12y x x a = 1 > 0 concave up c = 12 intercept is 0,12y
y
x
12
e.g. 2Graph 8 12y x x a = 1 > 0 concave up c = 12 intercept is 0,12y
zeroes y
x
12
e.g. 2Graph 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
e.g. 2Graph 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
e.g. 2Graph 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
e.g. 2Graph 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 2,0x
y
x
12
2 8 12y x x
e.g. 2Graph 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 2,0x
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
OR6 22
x
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
OR6 22
x
4
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
OR6 22
x
4
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
OR6 22
x
4
vertex
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
OR6 22
x
4
vertex 24 8 4 12y
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
OR6 22
x
4
vertex 24 8 4 12y
4
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
OR6 22
x
4
vertex 24 8 4 12y
4 vertex is 4, 4
y
x
12
–2 –6
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
OR6 22
x
4
vertex 24 8 4 12y
4 vertex is 4, 4
y
x
12
–2 –6
(–4, –4)
2 8 12y x x
e.g. 2Graph 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 2,0x
AOS2
bxa
824
OR6 22
x
4
vertex 24 8 4 12y
4 vertex is 4, 4
y
x
12
–2 –6
(–4, –4)
2 8 12y x x
(ii) Find the quadratic with;a) roots 3 and 6
(ii) Find the quadratic with;a) roots 3 and 6
2 9 18y a x x
(ii) Find the quadratic with;a) roots 3 and 6
2 9 18y a x x
6 3 6 3
(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
(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
25,3 : 3 5 10 5 16a
3 913
a
a
21 10 163
y x x
(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
(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)
(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
(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
25,3 : 3 5 10 5 16a
(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
25,3 : 3 5 10 5 16a
3 913
a
a
(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
25,3 : 3 5 10 5 16a
3 913
a
a
21 10 163
y x x
(iii) Solve;2) 5 6 0a x x
(iii) Solve;2) 5 6 0a x x
2 3 0x x
(iii) Solve;2) 5 6 0a x x
2 3 0x x
y
x–3 –2
(iii) Solve;2) 5 6 0a x x
2 3 0x x
y
x–3 –2
(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?
(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?
(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
(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
(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
(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
(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
(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
(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 theparabola below the x axis?
2 3 4 0x x
(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 theparabola below the x axis?
2 3 4 0x x
(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 theparabola below the x axis?
4 1x
2 3 4 0x x
Exercise 8A; 1adf, 2adf, 3bd, 4bd, 5c, 6ade, 7d, 9ace, 12c,13b, 14a