PreCalc H ­ Aim #69 Parabolas.notebook
Aim #69: How do we sketch the graphs of parabolas?
Conic Sections
Curves that result from the intersection
of a right circular cone and a plane
Parabola
A parabola is the collection of all points P in the plane that are the
same distance from a fixed point F as they are from a fixed line D.
The point F is called the focus of the parabola, and the line D is its
directrix.
As a result, a parabola is the set of points P for which
d(F, P) = d( P, D);
d = distance
Since the vertex is a point on
the parabola it must satisfy
the definition.
d(F,V) = d(V,D)
We will call this distance a.
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PreCalc H ­ Aim #69 Parabolas.notebook
Deriving the equation of a parabola where the focus is on the
positive x -axis.
d(F,P) = d(P,D)
x2 = 4py
y2 = 4px
y-axis symmetry
vertex: origin
p > 0 opens up
p < 0 opens down
focus: (0, p)
directrix: y = -p
x-axis symmetry
vertex: origin
p > 0 opens right
p < 0 opens left
focus: (p, 0)
directrix: x = -p
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PreCalc H ­ Aim #69 Parabolas.notebook
To sketch a parabola
1.
2.
3.
4.
5.
opens?
vertex
focus
directrix
endpoints of the latus rectum - half of 4p from the focus
eg. Sketch the parabola x2 = -8y
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PreCalc H ­ Aim #69 Parabolas.notebook
eg. Sketch the parabola y2 = 6x
Write the equation of a parabola with vertex at (0, 0)
and directrix the line x = 3/4.
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PreCalc H ­ Aim #69 Parabolas.notebook
Wrap it Up!
Write the equation of the parabola that is symmetric with
respect to the y-axis, has its vertex at the origin, and contains
the point (6,3).
x2 = 12y
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