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Table of Contents
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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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
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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
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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
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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
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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.
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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
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• 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
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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
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
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Summary:
Narrow
Face down
2,3V
2x Axis:
Point: 1,0
Function:2( ) 3 12 9f x x x
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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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