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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