8.1
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Copyright © 2011 Pearson, Inc.
8.1Parabolas
Goal: Find the equation, focus, and directrix of a parabola.
Copyright © 2011 Pearson, Inc. Slide 8.1 - 2
What you’ll learn about
Geometry of a Parabola Translations of Parabolas Reflective Property of a Parabola
… and whyConic 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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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 theparabola. The heat source is at the focus and heat is concentratedin parallel rays. Have you walked by the parabolic reflectorheater at COSTCO?
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Path of a Ball
Gallileo was the first to show that the path of an object thrown inspace is a parabola.
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Suspension Cables on the Golden Gate Bridge
Copyright © 2011 Pearson, Inc. Slide 8.1 - 8
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.
Copyright © 2011 Pearson, Inc. Slide 8.1 - 9
Graphs of x2 = 4py
Copyright © 2011 Pearson, Inc. Slide 8.1 - 10
Graphs of y2 = 4px
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Form of Equation
Vertex (0, 0) (0, 0)
Direction of Opening
Focus (0, p) (p, 0)
Directrix
Length of Focal Chord
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Form of Equation
Vertex (h, k) (h, k)
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For each parabola, find the vertex, focus, directrix, and focal chord length then sketch.
vertex: ____________focus: _____________directrix: _________focal chord: _________
𝑦=1
12 𝑥2
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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) anddirectrix of x = 5.
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Find the equation of the parabola and sketch its graph.
Focus at (3, -2) anddirectrix 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: __________
Copyright © 2011 Pearson, Inc. Slide 8.1 - 21
Example Finding an Equation of a Parabola
Find the standard form of the equation for the parabolawith vertex at (1,2) and focus at (1, 2).
Copyright © 2011 Pearson, Inc. Slide 8.1 - 22
Example Finding an Equation of a Parabola
Find an equation in standard form for the parabolawhose directrix is the line x 3 and whose focus isthe point ( 3,0).
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