Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 8.2 Parabolas and Modeling.

Copyright © 2013, 2009, 2005 Pearson Education, Inc.

Section 8.2

Parabolas and Modeling


Objectives

• Vertical and Horizontal Translations

• Vertex Form

• Modeling with Quadratic Functions (Optional)


Vertical and Horizontal Translations

The graph of y = x2 is a parabola opening upward with vertex (0, 0).

All three graphs have the same shape.y = x2

y = x2 + 1 shifted upward 1 unity = x2 – 2 shifted downward 2 units

Such shifts are called translations because they do not change the shape of the graph only its position



The graph of y = x2 is a parabola opening upward with vertex (0, 0).y = x2

y = (x – 1)2 Horizontal shift to the right 1 unit



The graph of y = x2 is a parabola opening upward with vertex (0, 0).y = x2

y = (x + 2)2 Horizontal shift to the left 2 units


Example

Sketch the graph of the equation and identify the vertex.

SolutionThe graph is similar to y = x2 except it has been translated 3 units down.

The vertex is (0, 3).

2 3y x


Example


SolutionThe graph is similar to y = x2 except it has been translated left 4 units.


2( 4)y x


Example


SolutionThe graph is similar to y = x2 except it has been translated down 2 units and right 1 unit.


2( 1) 2y x


Example

Compare the graph of y = f(x) to the graph of y = x2. Then sketch a graph of y = f(x) and y = x2 in the same xy-plane.

SolutionThe graph is translated to the right 2 units and upward 3 units.The vertex for f(x) is (2, 3) and the vertex of y = x2 is (0, 0).The graph opens upward and is wider.

21( ) ( 2) 3

4f x x


Example

Write the vertex form of the parabola with a = 3 and vertex (2, 1). Then express the equation in the form y = ax2 + bx + c.SolutionThe vertex form of the parabola is where the vertex is (h, k).a = 3, h = 2 and k = 1

To write the equation in y = ax2 + bx + c, do the following:

2( ) ,y a x h k

2)3( 12y x

2)3( 12y x 2( 4 43 1)y x x

23 12 12 1y x x 23 12 13y x x


Example

Write each equation in vertex form. Identify the vertex. a. b.Solutiona. Because , add and subtract 16 on the right.

2 8 13y x x 22 8 7y x x

2 28

162 2

b

2 8 13y x x

2 16 1 68 13y x x

24 3y x



Example (cont)

b. This equation is slightly different because the leading coefficient is 2 rather than 1. Start by factoring 2 from the first two terms on the right side.

22 8 7y x x

2 24

42 2

b

2

2

2 8 7

2( 4 ) 7

y x x

x x

2 42 4 74y x x

2 42 4 7 8y x x

22 2 1y x The vertex is ( 2, 1).

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 8.2 Parabolas and Modeling.

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Transcript of Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 8.2 Parabolas and Modeling.