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Section 9.3
The Parabola
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Definition of a Parabola
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Example
2
Find the vertex and the axis of symmetry of the parabola given
by y 2( 3) 4. Does it open up or down?x= − −
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Standard Form of the Equation of a Parabola
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Example
2
Find the focus and the directrix of the parabola given
by y 24 .x=
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Using the Standard Form of the Equation of a Parabola
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Example
2
Find the focus and the directrix of the parabola
given by x 4 .y= −
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Example
2
Find the focus and the directrix of the parabola
given by y 12 .x= −
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Translations of Parabola
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Example
Find the standard form of the equation of a parabola
with focus (0,3) and directrix of y=-1
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Example
2
Find the vertex, focus and directrix of the parabola
given by (x-3) 8( 1).y= +
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Example
2
Find the vertex, axis, focus, and directrix of the parabola
by completing the square on the equation
8y=2x 12 14.x− −
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Example
2
Find the vertex, axis, focus, and directrix of the parabola
by completing the square on the equation
6 16 25 0.y y x− + + =
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Applications
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Applications of Parabolas- cables on suspension bridges, arches on bridges, solar
cookers, reflectors on lights, satellite dishes
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Example
A student scientist wants to construct a solar cooker which is
in the shape of a parabolic dish with a diameter of 6 feet and a
depth of 1 foot. Where should the cooker (focus) be located
to get the maximum reflected sun rays to cook the food.
6 feet
1 foot
(3,1)
(0,0)
cooker
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Degenerate Conic Sections
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Intersections might now result in a conic section. Three degenerate cases occur
when the cutting plane passes through the
vertex. These degenerate conic sections are a point, line, and a pair of intersecting lines.
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(a)
(b)
(c)
(d)
2
Find the focus and directrix of the parabola with the given
equation. y 16 .x=
( 4,0), 4
(4,0), 4
(4,0), 4
(0, 4), 4
y
x
y
x
− = −
= −
=
=
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(a)
(b)
(c)
(d)
2
Find the vertex of the following parabola
(x+4) 12( 2).y= +
(2,4)
(4,2)
( 4, 2)
( 2, 4)
− −
− −
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(a)
(b)
(c)
(d)
2
Find the focus of the following parabola
(x+4) 12( 2).y= +
(2, 4)
(4,1)
( 4, 2)
( 4,1)
−
− −
−