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Graphing Quadratic Functions – Concept

• A quadratic function in what we will call Standard Form is given by:

2( ) ( )f x a x h k • The graph of a quadratic function is called a parabola. Here is the graph of a very simple quadratic function:

2( )f x x

4

2

-2

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• The value of the coefficient a determines the direction the parabola faces.

2( ) ( )f x a x h k

• When the value of a is positive, the parabola faces up.

• When the value of a is negative, the parabola faces down.

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• Example 1:

2( ) 3f x x 2( ) 3f x x0a 0a

-2

-4

4

2

Face Up

Face Down

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• The value of the coefficient a also determines the shape of the parabola.

2( ) ( )f x a x h k

• When |a| > 1 the parabola is narrow.

• When 0 < |a| < 1 the parabola is wide.

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• Example 2:

2( ) 3f x x 21( )

3f x x

1a 0 1a

4

2

4

2

-2NarrowWide

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• The vertex of a parabola is the highest point or the lowest point on the graph of a parabola.

6

4

2

5

(2,1)

2

-2

(-1,2)

VertexVertex

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• The vertex of a parabola whose function is given in standard form …

2( ) ( )f x a x h k

… is given by V(h,k).

• Example 3:2( ) 3( 2) 4f x x

The vertex is given by: (2,4)V

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• Example 3:

2( ) 2( 3) 5f x x

The vertex is given by: ( 3, 5)V

2( ) ( )f x a x h k

Put the function in the form of …

2( ) 2( ( 3)) ( 5)f x x

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2( ) 2( 3) 5f x x

The vertex is given by: ( 3, 5)V

Here is an easier way to work the last problem:

For the h value, take the opposite sign … For the k value, take

the same sign …

3h5k

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• Example 4:

22( ) ( 5) 7

3f x x

The vertex is given by: ( 5,7)V

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• The axis of symmetry of a parabola is the vertical line going through the vertex.

6

4

2(-1,2)

Notice the symmetry of the two branches of the parabola about the axis.

• Example 5:

Draw the axis

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6

4

2(-1,2)

• The equation of the axis of symmetry is given by

where h is the x-value of the vertex.x h

In this case, the equation of the axis of symmetry is given by:

1x

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SUMMARY

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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