Sect' 8'2 Parabolas

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Sect. 8.2 Parabolas
Goal 1 Write equations of Parabolas in
Standard Form. Goal 2 Graph Parabolas
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PARABOLAS
A Parabola is the set of points in the plane that
are equidistant from a fixed line (the Directrix)
and a fixed point (the Focus) not on the
directrix.
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The cross section of a headlight is an example of
a parabola...
The light source is the Focus
Directrix
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Some other applications of the parabola...
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d2
Focus
d1
d2
d3
d1
d3
Vertex
Directrix
Notice that the vertex is located at the midpoint
between the Focus and the Directrix...
Also, notice that the distance from the Focus to
any point on the parabola is equal to the
distance from that point to the Directrix...
We can determine the coordinates of the focus,
and the equation of the directrix, given the
equation of the parabola....
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Information About of Parabolas
Form of Equation
y a(x h)2 k
x a(y k)2 h
Vertex
(h, k)
(h, k)
Axis of Symmetry
x h
y k
Focus
Directrix
Right if a gt 0 Left if a lt 0
Direction of Opening
Upward if a gt 0 Downward if a lt 0
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Given the equation y - x2 2x 3
a) Write equation in Standard Form
y - (x 1)2 4
b) Identify the Vertex
(- 1, 4)
c) Identify the Axis of Symmetry
x - 1
d) Tell the Direction of Opening for the Parabola
Opens Downward
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Graph
y 2x2
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Graph
y 2(x 1)2 - 5
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Graph
x y2 4y - 1
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Determine the focus and directrix of the parabola
y 4x2

• Since x is squared, the parabola goes up or down
• Find Vertex (0, 0)

Lets see what this parabola looks like...
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Determine the focus and directrix of the
parabola 3y2 12x 0
Standard Form

• Since y is squared, the parabola goes left or
  right
• Find Vertex (0, 0)

Focus (1, 0) Directrix x 1
Lets see what this parabola looks like...
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Write an equation for Parabola described. Then
Graph.
Focus (3, 8) Directrix y 4
(3, 8)
Find Vertex (h, k)
Vertex (3, 6)
Find value of a
y 4
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Write an equation for Parabola described. Then
Graph.
Focus (3, - 1) Vertex (5, - 1)
Find Directrix
x 7
Find the value of a
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Analyzing a Parabola
Find the coordinates of the vertex and focus, the
equation of the directrix, the axis of symmetry,
and the direction of opening of y2 - 8x - 2y - 15
0.
y2 - 8x - 2y - 15 0 y2 - 2y _____ 8x
15 _____
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(y - 1)2 8x 16 (y - 1)2 8(x 2)
Standard form
The vertex is (-2, 1). The focus is (0, 1). The
equation of the directrix is x - 4. The axis of
symmetry is y 1. The parabola opens to the
right.
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Write an equation of the parabola whose vertex is
at (2, 1) and whose focus is at (3, 1).
SOLUTION
Begin by sketching the parabola. The parabola
opens to the left.
Find h and k The vertex is at (2, 1), so h
2 and k 1.
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Write an equation of the parabola whose vertex is
at (2, 1) and whose focus is at (3, 1).
SOLUTION
(2, 1)
Find the value of a.
x a(y k) 2 h
The standard form of the equation is