Goal: Find the equation, focus, and directrix of a parabola.
8.1
Parabolas
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What you’ll learn about
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Geometry of a Parabola
Translations of Parabolas
Reflective Property of a Parabola
… and why
Conic sections are the paths of nature: Any free-moving
object in a gravitational field follows the path of a conic
section.
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Slide 8.1 - 2
Golden Arches
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Satellite Dishes
Incoming Waves are concentrated to the focus.
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Heaters
Heaters are sold which make use of the reflective property of the
parabola. The heat source is at the focus and heat is concentrated
in parallel rays. Have you walked by the parabolic reflector
heater at COSTCO?
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Path of a Ball
Gallileo was the first to show that the path of an object thrown in
space is a parabola.
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Suspension Cables on the Golden Gate
Bridge
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Parabola
A parabola is the
set of all points in
a plane equidistant
from a particular
line (the directrix)
and a particular
point (the focus)
in the plane.
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Slide 8.1 - 8
Graphs of x2 = 4py
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Slide 8.1 - 9
Graphs of y2 = 4px
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Form of Equation
Vertex
𝟏 𝟐
𝒚=
𝒙
𝟒𝒑
𝟏 𝟐
𝒙=
𝒚
𝟒𝒑
(0, 0)
(0, 0)
(0, p)
(p, 0)
𝑦 = −𝑝
𝑥 = −𝑝
Direction of
Opening
Focus
Directrix
Length of Focal
Chord
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4𝑝
4𝑝
Form of Equation
𝟏
𝒚−𝒌=
𝒙−𝒉
𝟒𝒑
𝒙−𝒉
Vertex
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𝟐
𝟐
= 𝟒𝒑 𝒚 − 𝒌
(h, k)
𝟏
𝒙−𝒉=
𝒚−𝒌
𝟒𝒑
𝒚−𝒌
𝟐
𝟐
= 𝟒𝒑 𝒙 − 𝒉
(h, k)
For each parabola, find the vertex, focus,
directrix, and focal chord length then sketch.
1 2
𝑦=
𝑥
12
vertex: ____________
focus: _____________
directrix: _________
focal chord: _________
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vertex: ____________
focus: _____________
directrix: _________
focal chord: _________
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vertex: ____________
focus: _____________
directrix: _________
focal chord: __________
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vertex: ____________
focus: _____________
directrix: _________
focal chord: __________
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vertex: ____________
focus: _____________
directrix: ________
focal chord: ________
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Find the equation of the parabola and
sketch its graph.
Vertex at (0, 0) and
directrix of x = 5.
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Find the equation of the parabola and
sketch its graph.
Focus at (3, -2) and
directrix of y = 4.
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Find the vertex, focus, and length of the
focal chord for the parabola below.
vertex: _____________
focus: ______________
focal chord: __________
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Example Finding an Equation of a
Parabola
Find the standard form of the equation for the parabola
with vertex at (1,2) and focus at (1,  2).
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Slide 8.1 - 21
Example Finding an Equation of a
Parabola
Find an equation in standard form for the parabola
whose directrix is the line x  3 and whose focus is
the point (  3,0).
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Slide 8.1 - 22