11 x1 t10 01 graphing quadratics (2012)

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