f(x) = x2

Roots, Zeros, x-intercepts:

x=0

(0,0)

Vertex:

(0,0)

a Minimum

Axis of Symmetry:

the line

x=0

x f(x)

-3 9

-2 4

-1 1

0 0

1 1

2 4

3 9

a=1, c=0

Opens up;

PARENT

Quadratic

Key

f(x) = 2x2-4x-6

=2(x+1)(x-3)

Roots, Zeros, x-intercepts:

x=-1, 3

(-1,0), (3,0)

Vertex:

(1,-8)

a Minimum

Axis of Symmetry:

the line

x=1

x f(x)

-2 10

-1 0

0 -6

1 -8

2 -6

3 0

4 10

a>1, c<0

Opens up;

Shifted down

and narrower

than

PARENT

Quadratic

Key

f(x) =

x2/3+2x/3+4/3

=(1/3)(x2+2x+4)

Roots, Zeros, x-intercepts:

NONE

Vertex:

(-1,1)

a Minimum

Axis of Symmetry:

the line

x=-1

x f(x)

-10 28

-7 13

-4 4

-1 1

2 4

5 13

8 28

0<a<1, c>0

Opens up;

Shifted up and

wider than

PARENT

Quadratic

Key

f(x) = -x2-4x+5

= -1•(x2+4x-5)

= -(x+5)(x-1)

Roots, Zeros, x-intercepts:

X=-5, 1

(-5,0), (1,0)

Vertex:

(-2,9)

a Maximum

Axis of Symmetry:

the line

x=-2

x f(x)

-5 0

-4 5

-3 8

-2 9

-1 8

0 5

1 0

a<0 (a=-1), c>0

Opens down;

Flipped, shifted

up, and same

width as

PARENT

Quadratic

Key

f(x) = -½x2+2x+6

= -½(x2-4x-12)

= -½(x-6)(x+2)

Roots, Zeros, x-intercepts:

X = 6, -2

(6,0), (-2,0)

Vertex:

(2,8)

a Maximum

Axis of Symmetry:

the line

x=2

x f(x)

-4 -10

-2 0

0 6

2 8

4 6

6 0

8 -10

-1<a<0, c>0

Opens down;

Flipped, shifted

up, and wider

than

PARENT

Quadratic

Key

f(x)=-2x2+12x-20

= -2(x2-6x+10)

Roots, Zeros, x-intercepts:

NONE

Vertex:

(3,-2)

a Maximum

Axis of Symmetry:

the line

x=3

x f(x)

0 -20

1 -10

2 -4

3 -2

4 -4

5 -10

6 -20

a<-1, c<0

Opens down;

Flipped, shifted

down, and

narrower than

PARENT

Quadratic

Key

f(x) = 4x2-4

= 4(x2-1)

= 4(x+1)(x-1)

Roots, Zeros, x-intercepts:

x = 1, -1

(1,0), (-1,0)

Vertex:

(0,-4)

a Minimum

Axis of Symmetry:

the line

x=0

x f(x)

-3 32

-2 12

-1 0

0 -4

1 0

2 12

3 32

a>1, c<0

Opens up;

Shifted down

and narrower

than

PARENT

Quadratic

Key

f(x) = -¼x2+4

= -¼(x2-16)

= -¼(x+4)(x-4)

Roots, Zeros, x-intercepts:

x = 4, -4

(4,0), (-4,0)

Vertex:

(0,4)

a Maximum

Axis of Symmetry:

the line

x=0

x f(x)

-6 -5

-4 0

-2 3

0 4

2 3

4 0

6 -5

-1<a<0, c>0

Opens down;

Flipped, shifted

up and wider

than

PARENT

Quadratic

Key

f(x) = ½x2+2x+2

= ½(x2+4x+4)

= ½(x+2)(x+2)

= ½(x+2)2

Roots, Zeros, x-intercepts:

x = -2

(-2,0)

Vertex:

(-2,0)

a Minimum

Axis of Symmetry:

the line

x=-2

x f(x)

-8 18

-6 8

-4 2

-2 0

0 2

2 8

4 18

0<a<1, c>0

Opens up;

c>0, not shifted

up! Why?

Wider than

PARENT

Quadratic

Key

f(x) = -x2+6x-9

= -1•(x2-6x+9)

= -(x-3)(x-3)

= -(x-3)2

Roots, Zeros, x-intercepts:

x = 3

(3,0)

Vertex:

(3,0)

a Maximum

Axis of Symmetry:

the line

x=3

x f(x)

0 -9

1 -4

2 -1

3 0

4 -1

5 -4

6 -9

a<0 (a=-1), c>0

Opens down;

c>0, not shifted

up! Why?

Same width as

PARENT

Quadratic

Key

f(x) = ½x2+2x+2

= ½(x2+4x+4)

= ½(x+2)(x+2)

= ½(x+2)2

f(x) = x2

f(x) = -x2+6x-9

= -1•(x2-6x+9)

= -(x-3)(x-3)

= -(x-3)2

f(x) = -¼x2+4

= -¼(x2-16)

= -¼(x+4)(x-4)

x f(x)

-8 18

-6 8

-4 2

-2 0

0 2

2 8

4 18

0<a<1, c>0

Opens up;

c>0, not shifted

up! Why?

Wider than

PARENT

Quadratic

Key

f(x) =

x2/3+2x/3+4/3

=(1/3)(x2+2x+4)

f(x) = -¼x2+4

= -¼(x2-16)

= -¼(x+4)(x-4)

f(x) = -½x2+2x+6

= -½(x2-4x-12)

= -½(x-6)(x+2)

f(x) = 2x2-4x-6

=2(x+1)(x-3)

f(x)=-2x2+12x-20

= -2(x2-6x+10)

f(x) = x2

f(x) = 4x2-4

= 4(x2-1)

= 4(x+1)(x-1)

f(x) = -x2-4x+5

= -1•(x2+4x-5)

= -(x+5)(x-1)

Roots, Zeros, x-intercepts:

x=-1, 3

(-1,0), (3,0)

Roots, Zeros, x-intercepts:

x = 4, -4

(4,0), (-4,0)

Roots, Zeros, x-intercepts:

X=-5, 1

(-5,0), (1,0)

Roots, Zeros, x-intercepts:

NONE

Roots, Zeros, x-intercepts:

X = 6, -2

(6,0), (-2,0)

Roots, Zeros, x-intercepts:

NONE

Roots, Zeros, x-intercepts:

x = 1, -1

(1,0), (-1,0)

Roots, Zeros, x-intercepts:

x=0

(0,0)

Vertex:

(3,-2)

a Maximum

Vertex:

(-1,1)

a Minimum

Vertex:

(0,0)

a Minimum

Vertex:

(0,4)

a Maximum

Vertex:

(0,-4)

a Minimum

Vertex:

(1,-8)

a Minimum

Vertex:

(-2,9)

a Maximum

Vertex:

(2,8)

a Maximum

Axis of Symmetry:

the line

x=3

Axis of Symmetry:

the line

x=-1

Axis of Symmetry:

the line

x=2

Axis of Symmetry:

the line

x=0

Axis of Symmetry:

the line

x=0

Axis of Symmetry:

the line

x=1

Axis of Symmetry:

the line

x=-2

Axis of Symmetry:

the line

x=0

x f(x)

-6 -5

-4 0

-2 3

0 4

2 3

4 0

6 -5

x f(x)

-5 0

-4 5

-3 8

-2 9

-1 8

0 5

1 0

x f(x)

-3 32

-2 12

-1 0

0 -4

1 0

2 12

3 32

x f(x)

-2 10

-1 0

0 -6

1 -8

2 -6

3 0

4 10

x f(x)

-3 9

-2 4

-1 1

0 0

1 1

2 4

3 9

x f(x)

0 -20

1 -10

2 -4

3 -2

4 -4

5 -10

6 -20

x f(x)

-10 28

-7 13

-4 4

-1 1

2 4

5 13

8 28

x f(x)

-4 -10

-2 0

0 6

2 8

4 6

6 0

8 -10

a<0 (a=-1), c>0

Opens down;

Flipped, shifted

up, and same

width as

PARENT

Quadratic

a>1, c<0

Opens up;

Shifted down

and narrower

than

PARENT

Quadratic

a>1, c<0

Opens up;

Shifted down

and narrower

than

PARENT

Quadratic

a<-1, c<0

Opens down;

Flipped, shifted

down, and

narrower than

PARENT

Quadratic -1<a<0, c>0

Opens down;

Flipped, shifted

up and wider

than

PARENT

Quadratic

-1<a<0, c>0

Opens down;

Flipped, shifted

up, and wider

than

PARENT

Quadratic

a=1, c=0

Opens up;

PARENT

Quadratic

0<a<1, c>0

Opens up;

Shifted up and

wider than

PARENT

Quadratic