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Sections 9.4 and 9.5 
Conics and Calculus
Parabolas
—A parabola is the set of all points (
x, y
) in the plane that are equidistant from a fixed line (directrix) and a fixed
point (focus).
The Standard Form of the equation of a parabola with vertex (
h, k
) and directrix
p
k
y
is
)
(
4
)
(
2
k
y
p
h
x
For this type, the axis is vertical (i.e. the parabola opens up or down)
For a directrix
p
h
x
, the equation is
)
(
4
)
(
2
h
x
p
k
y
.
Here the axis is horizontal, or the parabola opens to the side
Note:
For parabolas
, the vertex is always p units from both the focus and the directrix
.
If you know two pieces of
information from among vertex, focus, directrix, p, you can find everything there is to know about a parabola.
Also note that general form for a parabola is
c
bx
ax
y
2
(opens up or down) or
c
by
ay
x
2
(opens sideways).
When given in this form
p
a
4
1
and
you can find the vertex using either College Algebra or Calc I techniques
For each of the examples below, find the vertex, focus, and directrix of the parabola, and sketch the graph.
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 Spring '12
 PamelaSatterfield
 Calculus

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