Section 6-5:  Parabolas
 
 
A parabola is the set of all points P in the plane that are equidistant from a fixed point F (focus) and a fixed line d (directrix).
Demonstration of Focus Point for a Parabola (Manipula Math)
Drawing a Parabola (Manipula Math)

 
The equations of the parabola are as follows:
For parabolas opening up/down, the directrix is a horizontal line in the form y = + p
For parabolas opening right/left, the directrix is a vertical line in the form
x = + p
The vertex point for all of the above is (0, 0)

 Sample Problems
 
1)  Find the focus point and directrix and graph the parabola:  y = x2/8
 
                 Solution:  The parabola opens up.  1/4p = 1/8  means 4p = 8 and
p = 2.  This is the distance from the vertex to the directrix or to the focus point.  The focus point is 2 units up so it is (0, 2).  The directrix is a horizontal line 2 units down from the vertex.  The equation is y = -2
To determine how wide the parabola opens, the distance |4p| is the distance of the chord connecting the two sides of the parabola through the focus point perpendicular to the axis of symmetry.  In this case 4p = 8, so the parabola is 4 units from the focus point both right and left.  The two points are given on the graph.
 
2)  Find the focus point and the directrix and graph the parabola:  x = -2y2
 
                 Solution:  This parabola opens to the left.  1/4p = -2
 -8p = 1
p = -1/8
The focus point is at (-1/8, 0) and the directrix is a vertical line at x = 1/8
The distance across the parabola through the focus is 1/2, so the parabola is one-fourth unit up and down from the focus point.
 
3)  Find the equation of the parabola with vertex at (0, 0) and directrix
y = 2.
                 Solution:  Since the directrix is a horizontal line and is above the vertex, the parabola opens down.  p = 2 (distance from directrix to vertex), so 4p = 8.  Thus the equation is y = -(1/8)x2
 
4)  Find the equation of a parabola with focus at (2, 0) and directrix at x = -2
                 Solution:  The vertex for this parabola is inbetween the directrix and focus.  So the vertex is (0, 0).  The parabola opens to the right with p = 2.
So 4p = 8.  Thus the equation is x = (1/8)y2

Translations of the parabola
 
The equations of the parabola with vertex (h, k) are:

 
5)  Find the vertex, focus and directrix and graph the parabola
y = 2x2 - 8x + 1
Solution:
Put the equation in the correct form.
y - 1 = 2(x2 - 4x     ) Complete the square
y - 1 + 8 = 2(x2 - 4x + 4)  added 8 to both sides!
y + 7 = 2(x - 2)2
The parabola opens up with vertex at (2, -7)
1/4p = 2
8p = 1
p = 1/8
Focus point at (2, -6 7/8)
directrix at y = -7 1/8
 
6)  Find the equation of the parabola with focus ( 1, 3) and directrix x = -3.
Solution:
The parabola opens to the right.  The vertex is midway between the focus and directrix.  The vertex is at (-1, 3).  p = 2 so 4p = 8
The equation is:
(x + 1) = (1/8)(y - 3)2

 
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