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,
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 =
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,
5) Find the vertex, focus and directrix
and graph the parabola
y = 2x2
- 8x + 1
Put the equation in the
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
directrix at y = -7 1/8
6) Find the equation of the parabola
with focus ( 1, 3) and directrix x = -3.
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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