Graphing Quadratic Functions – General Form

• It is assumed that you have already viewed the previous three slide shows titled

Graphing Quadratic Functions – ConceptGraphing Quadratic Functions – Standard FormGraphing Quadratic Functions – Converting

General Form To Standard Form

• The next three pages are summaries from those shows, and are important to understanding this module.

2( )f x ax bx c

General Form of a Quadratic Function

Standard Form of a Quadratic Function

2( )f x a x h k

Graphing a Quadratic Function in Standard Form

0a Face Up

0a Face Down

1a Narrow

0 1a Wide

2( ) ( )f x a x h k

Vertex ( , )V h k

Axis of symmetry

x h

1a

4

2

Narrow

4

2

-2

Wide

0 1a Plotting an Extra Point when Graphing Quadratics

Choose a value for x 1 unit away

from the vertex.

Choose a value for x more than 1 unit

away from the vertex

Vertex of the Graph of a Quadratic Function in General Form

2

bx

a

2( )f x ax bx c

• Given the general form of a quadratic function …

…the x-value of the vertex is given by

2

bx

a

• There are two other helpful concepts for graphing a quadratic function that is in general form.

We already know that the x-value of the vertex is given by …

• Concept 1:

The y-value of the vertex is given by …

2

bf

a

Thus, the vertex of the graph of the quadratic function in general form is given by

• Concept 2:

,2 2

b bV f

a a

The a of the general form is the same value as the a in the standard form.

This means that the a in the general form can be used to determine the face up/face down and narrow/wide behavior, just as it did in the standard form.

Summary for Graphing Quadratics in General Form

0a Face Up

0a Face Down

1a Narrow

0 1a Wide

2( )f x ax bx c

Vertex

Axis of symmetry

2

bx

a

,2 2

b bV f

a a

• Example:

Sketch the graph of the given quadratic function.

2( ) 3 12 9f x x x

3a Face down

3 1a Narrow

Vertex:2

bx

a

12

2 3

12

6

2

2( ) 3 12 9f x x x 2x

2( 2) 3 2 12 2 9f

3 4 12 2 9

12 24 9 3Vertex 2,3V

Axis of Symmetry: x = x-value of vertex

2x

Point: since the parabola is narrow, use an x-value that is only one unit from the vertex.

1x

2( 1) 3 1 12 1 9f

3 12 9 3 1 12 1 9

0 1,0

Summary:

Narrow

Face down

2,3V

2x Axis:

Point: 1,0

Function:2( ) 3 12 9f x x x

Plot the vertex 2,3V

2x Plot the axis:

Plot the point: 1,0

-5

4

2

-2

-4

-5

4

2

-2

-4

-5

4

2

-2

-4

Draw the branch of the parabola on the right side of the axis. -5

4

2

-2

-4

Use symmetry to draw the left branch.

-5

4

2

-2

-4Label

-5

4

2

-2

-4

(-1,0)

x=-2

(-2,3)