Section 10.1 Parabolas
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Transcript of Section 10.1 Parabolas
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Section 10.1 Parabolas
Objectives:•To define parabolas geometrically.•Use the equation of parabolas to find relevant information.•To find the equation of parabolas given certain information
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Parabola—Geometric Definition• A parabola is the set of points in the plane
equidistant from a fixed point F (focus) and a fixed line l (directrix).
– The vertex V lies halfway between the focus and the directrix.
– The axis of symmetry is the line that runs through the focus perpendicular to the directrix.
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Parabola with Vertical Axis of SymmetryThe graph of the equation y = ax2 is a
parabola with these properties. • vertex: V(0,0)• focus: F(0, p) where p is the distance
between the focus and vertex• directrix: y = -p • a = 1
4p(recall: a is the number that determines how wide or narrow the parabola is)
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Parabola with Vertical Axis• The parabola opens:
– Upward if p > 0.– Downward if p < 0.
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Ex 1. Find the equation of the parabola with vertex V(0,0) and focus F(0,2).
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Ex 2. Find the equation of the parabola with vertex (0,0) and directrix y = 4.
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Class Work 1. Find the equation of the parabola with focus
(0,-5) and vertex (0,0).
2. Find the equation of the parabola with focus (0,3) and directrix y = -3
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The equation of the parabola whose vertex is (h,k) is
Parabolas whose vertex is not at the origin:
2( )y a x h k
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Ex 3. Find the equation of the parabola with focus (3, -1) and vertex (3, -4).
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Ex. 4 Find the equation of the parabola with focus (4, -1) and vertex (4, 1).
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Class Work
3. Find the equation of the parabola with vertex (2, 8) and focus (2, 3).
4. Find the equation of the parabola with focus (-1, -3) and vertex (-1, 1)
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Parabola with Horizontal AxisThe graph of the equation x=ay2
is a parabola with these properties:
• Vertex: V(0,0)
• Focus: F(p, 0)
• directrix: x = -p
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Parabola with Horizontal Axis• The parabola opens:
– To the right if p > 0.– To the left if p < 0.
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Ex 5. Find the equation of the parabola with focus (6, 0) and vertex (0, 0).
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Class Work
5. Find the equation of the parabola with focus (-3,0) and directrix x = 3
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Parabolas whose vertex is not at the origin:
The equation of the parabola whose vertex is (h,k) is
2( )x a y k h
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Ex 6. Find the equation of the parabola with vertex (4, -2) and focus (2, -2)
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Class Work6. Find the equation of a parabola with focus
(3,2) and vertex (5, 2).
7. Find the equation of a parabola with vertex (-4,1) and directrix x = -7.
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HW#1 Parabolas Worksheet