Algebra 2

Chapter 9 Conic Sections: Circles and Parabolas

9-3 Parabolas

WARMUP: Determine the distance between the given

point and line. Draw a sketch if necessary.

1. ( 3, 4 ); x-axis

2. ( -1, 2 ); y-axis

3. ( -2, 3 ); x = 1

4. ( 5, -4 ); y = -2

5. ( 1, -3 ); x = -4

9-3 Parabolas

OBJECTIVE: To learn the relationships among the focus, directrix, vertex, and axis of a parabola and the equation of a parabola.

9-3 Parabolas

Cool Parabola sites: http://

www.ies.co.jp/math/java/conics/focus/focus.html

http://www.mathwarehouse.com/quadratic/parabola/interactive-parabola.php

http://www.ies.co.jp/math/java/conics/draw_parabola/draw_parabola.html

9-3 Parabolas

A new definition for a parabola:

A parabola is the set of all points equidistant from a fixed line, called the directrix, and a fixed point not on the line, called the focus.

9-3 Parabolas

Look closer:

9-3 Parabolas

IMPORTANT! The distance between the focus and the

vertex (call it c) is the same as the distance between the vertex and the directrix!

The parabola ALWAYS opens away from the directrix, and around the focus!!!

9-3 Parabolas

Typical problem at this stage: The vertex of a parabola is ( -5, 1 ) and the

directrix is the line y = -2. Find the focus of the parabola.

9-3 Parabolas

The equation of a parabola is:

where ( h, k ) is the vertex of the parabola, and a determines how the curve opens, and a basic shape.

2( ) ( )y k a x h

9-3 Parabolas

What about a parabola that looks like this?

9-3 Parabolas

Let’s just get right to it: A parabola that opens left or right will have

an equation in the form:

What is different? Is this a function?

2( ) ( )x h a y k

9-3 Parabolas

Some basics: If a>0, the parabola will open to the right. If a<0 the parabola will open to the left. ( h, k ) is still the vertex, as always. The axis of symmetry will be y=k. The directrix will be x=?.

2( ) ( )x h a y k

9-3 Parabolas

Look at example 2 in the book on page 413.

9-3 Parabolas

IMPORTANT!!!

If the distance between the vertex and the focus of the parabola is |c|, then it can be shown

that in the equation of the

parabola.

1

4a

c

9-3 Parabolas

The parabola whose equation is

opens upward if a>0, downward if a<0

has vertex V( h, k )

focus F( h, k + c )

directrix y = k – c

and axis of symmetry x = h.

2 1( ) ( ) where

4y k a x h a

c

9-3 Parabolas

The parabola whose equation is

opens to the right if a>0, to the left if a<0

has vertex V( h, k )

focus F( h + c, k )

directrix x = h – c

and axis of symmetry y = k.

2 1( ) ( ) where

4x h a y k a

c

9-3 Parabolas

STEPS TO SOLVE!!!

1. ALWAYS - Draw a picture with the info you are given! THIS WILL HELP!!

2. From the picture, determine which way your parabola will open. Roughly sketch it.

3. Determine the value of c.

4. Determine a.

5. Write your equation and all the pieces.

9-3 Parabolas

Let’s look at some problems:

9-3 Parabolas

9-3 Parabolas

9-3 Parabolas