Section 9.2 The Parabola
Recall that a parabola can be obtained as the graph of the quadratic equation
Ax
2
+
Bxy
+
Cy
2
+
Dx
+
Ey
+
F
= 0 with discriminant
B
2

4
AC
= 0. An alternate method for obtaining a
parabola is given by the following deﬁnition.
Deﬁnition.
Let
D
(
directrix
) be any line and
F
(
focus
)anypo
intnoton
D
. The set of all points
P
such that
d
(
P,F
)=
d
(
P,D
) is called a
parabola
. (Note:
d
(
A, B
) denotes the distance from
A
to
B
.)
The formula of a parabola is simpliﬁed if the parabola’s directrix is either vertical or horizontal.
The equations and graphs of parabolas with vertical and horizontal directrixes and vertices at the
origin are shown below.
Example.
Graph and discuss the equation
y
2
=12
x
.
Solution. Since
y
2
x
is a quadratic equation with discriminant equal to zero,
y
2
x
is the
equation of a parabola. Comparing the equation
y
2
x
to the equations given above indicates
1
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View Full Documentthat the parabola has its vertex at the origin. Since
y
can assume both positive and negative values
and
x
can assume only positive values, the parabola opens to the right. Comparing
y
2
=12
x
to
y
2
=4
ax
,weseethat4
a
=12and
a
= 3. Since
a
= 3, the focus of the parabola is at
F
=(3
,
0)
and the directrix is the line
x
=

3. A graph of the parabola is shown below.
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 Fall '11
 Kutter
 Algebra, Conic section

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