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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.

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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

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2( ) ( )f x a x h k

Sketch the Graph of a Quadratic in Standard Form

0a Face Up

0a Face Down

Vertex ( , )V h k

Axis of symmetry

x h

x-int: f (x) = 0 and solve for x

y-int: x = 0 and solve for y

Draw the parabola

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Vertex - General Form

2bxa

2( )f x ax bx c

• Given the general form of a quadratic function …

…the x-value of the vertex is given by

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2bxa

• 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:

To find the y-value of the vertex, just plug the x-value into the function …

2bfa

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

• Concept 2:

,2 2b bV fa 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 behavior, just as it did in the standard form.

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Summary for Graphing Quadratics in General Form

0a Face Up

0a Face Down

2( )f x ax bx c

Axis of symmetry 2

bx

a

,2 2b b

V fa a

x-int: f (x) = 0 and solve for x

y-int: x = 0 and solve for y

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• Example:

Sketch the graph of the given quadratic function.

2( ) 3 12 9f x x x

3a Face down

Vertex:2bxa

12

2 3

126

2

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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

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Axis of Symmetry: x = x-value of vertex

2x

2,3V

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x-intercepts

2( ) 3 12 9f x x x

2

2

3 12 9 0

3 4 3 0

3 1 3 0

x x

x x

x x

3, 1x

3,0 1,0

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y-intercept

2( ) 3 12 9f x x x

0, 9

2(0) 3 0 12 0

9

f

0x

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Summary:

Face down

2,3V

2x Axis:

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

x-intercepts

3,0 1,0

y-intercept

0, 9

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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)

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