Algebra 2
Chapter 9 Conic Sections: Circles and
Parabolas
9-3 Parabolas
 WARMUP:
 Determine the distance between the given
1.
2.
3.
4.
5.
point and line. Draw a sketch if necessary.
( 3, 4 ); x-axis
( -1, 2 ); y-axis
( -2, 3 ); x = 1
( 5, -4 ); y = -2
( 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/f
ocus.html
 http://www.mathwarehouse.com/quadratic/p
arabola/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:
( y  k )  a ( x  h) 2
where ( h, k ) is the vertex of the parabola, and
a determines how the curve opens, and a
basic shape.
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:
( x  h)  a ( y  k )
 What is different?
 Is this a function?
2
9-3 Parabolas
( x  h)  a ( y  k )
2
 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=?.
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
1
a
4c
parabola.
in the equation of the
9-3 Parabolas
 The parabola whose equation is
1
( y  k )  a ( x  h) where a 
4c
2
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.
9-3 Parabolas
 The parabola whose equation is
1
( x  h)  a ( y  k ) where a 
4c
2
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.
9-3 Parabolas
 STEPS TO SOLVE!!!
1. ALWAYS - Draw a picture with the info you
2.
3.
4.
5.
are given! THIS WILL HELP!!
From the picture, determine which way your
parabola will open. Roughly sketch it.
Determine the value of c.
Determine a.
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