The Parabola

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Section 9.3

• The Parabola

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Finally, something familiar!

• The parabola is oft discussed in MTH 112, as it
  is the graph of a quadratic function
• Does look familiar?
• Our discussion of the parabola will be consistent
  with our discussion of the other conic sections.

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The Parabola

• A parabola is the set of all points in the plane
  that are the same distance from a given point
  (focus) as they are from a given line
  (directrix).
• It is important to note that the focus is not a
  point on the directrix.
• While the directrix can be any line, we only
  consider horizontal and vertical ones.

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Parabolic Parts

• The axis of symmetry of a parabola is an
  imaginary line perpendicular to the directrix
  that passes through the focus.
• The axis of symmetry intersects the parabola at a
  point called the vertex.
• Let p be the distance from the vertex to the
  focus. It follows that p is also the distance
  from the vertex to the directrix.

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More Pictures
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Equations

• The standard form of the equation of a parabola
  with horizontal directrix is
• When p is positive, the parabola opens upward.
• When p is negative, the parabola opens downward.

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Equations (cont.)

• The standard form of the equation of a parabola
  with vertical directrix is
• When p is positive, the parabola opens to the
  right.
• When p is negative, the parabola opens to the
  left.

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Giggle, giggle

• The latus rectum is a line segment that
• passes through the focus
• Is parallel to the directrix
• Has its endpoints on the parabola.
• The length of the latus rectum is 4p.

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Pictures
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Finally

• Draw the picture.
• Draw the picture.
• Draw the picture.
• And, when all else fails.
• Draw the picture.

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Examples

• Find the vertex, focus and directrix, and sketch
  the graph.
• x2 24y
• y2 40x
• (x 2)2 -4(y 1)
• Find the standard form of the equation the
  parabola so described
• Focus is (12, 0) directrix is x -12
• Vertex is (3, -1) focus is (3, -2)

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More Examples

• Convert to standard form by completing the square
  on x
• x2 6x 12y 15 0
• Convert to standard form by completing the square
  on y
• y2 12y 16x 36 0