Honors Math 2
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Geometric (Locus) Definition of a Parabola
A parabola is a locus defined in terms of a fixed point, called the focus, and a fixed line, called the
directrix. A parabola is the set of all points P(x, y) whose distance from the focus (F) equals its
distance from the directrix. In other words, PF = PD (where D is the point on the directrix closest to
P – so PD is perpendicular to the directrix).
The figure shows a parabola with its focus at
(0, 1) and a directrix of y  1 . Equally
spaced concentric circles with their center at
the parabola’s focus enable you to measure
distances from the focus. Equally spaced
 parallel to the parabola’s
horizontal lines
directrix enable you to measure vertical
distances from the directrix. This type of
graph paper is called focus-directrix graph
paper.
Notice that P is the point of intersection of
the circle centered at (0, 1) with a radius of 6
and the horizontal line 6 units above the
directrix. Thus, P is equidistant from the
focus and directrix. Examine the figure and
note that all points on the parabola are
equidistant from the focus and the directrix.
Example:
In the diagram, a focus has been
drawn at the point (0, 3) and a
directrix has been drawn, y = –3.
a. Plot the points that are
equidistant from the focus and
directrix using the concentric circles
and the grid.
b. Identify the vertex of the
resulting parabola.
c. Identify p, the distance from the
focus to the vertex (and the distance
from the directrix to the vertex).
Deriving the Algebraic Definition of a Parabola
For now, let’s look at parabolas whose focus is the point (0, p) and whose directrix is y = – p. This will mean
the vertex of the parabola is at the origin, as shown in the sketch below.
If PF = PD, derive the standard equation for any point P on this
parabola using the distance formula.
What if the directrix is a vertical line rather than a horizontal one?
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Example: Graph x   y 2 and identify the vertex, focus and directrix.
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
Example: Write the standard equation of a parabola with its vertex at the origin and with the directrix y = 4.
For the following three problems:
a. Plot the points that are equidistant from the focus and directrix using the concentric circles and the grid.
b. Identify the vertex of the resulting parabola.
c. Identify p, the distance from the focus to the vertex (and the distance from the directrix to the vertex).
d. Write the equation of the parabola.
1. Focus at (0, –3)
Directrix of y = 3
2. Focus at (3, 0)
Directrix of x = –3
3. Focus at (–3, 0)
Directrix of x = 3
For the next three problems: Write the standard equation for the parabola with the given characteristics.
4. Given its graph below.
5. Given its vertex at (0, 0)
6. Given its vertex at (0, 0)
and its focus at (0, -5)
and its directrix x = 2
7. Graph x 

1 2
y and identify the vertex, focus and directrix.
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