Today%20in%20Precalculus

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Today in Precalculus

• Notes Conic Sections - Parabolas
• Homework
• Test grades will be in home access before I leave
  today. Well go over them on Monday.

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Conic Sections

• Conic sections are formed by the intersection of
  a plane and a cone.

hyperbola
circle
ellipse
parabola
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Degenerate Conic Sections

• Atypical conics
• The conic sections can be defined algebraically
  in the Cartesian plane as the graphs of
  second-degree equations in two variables, that
  is, equations of the form Ax2 Bxy Cy2 Dx
  Ey F 0, where A, B, and C are not all zero

intersecting lines
point
line
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Parabolas

• Definition A parabola is the set of all points
  in a plane equidistant from the directrix and the
  focus in the plane.
• The line passing through the focus and
  perpendicular to the directrix is the axis of the
  parabola and is the line of symmetry for the
  parabola.
• The vertex is midway between the focus and the
  directrix and is the point of the parabola
  closest to both.

axis
focus
vertex
directrix
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P(x,y)

• By definition, the distance between F
  and P has to equal the distance between P
  and D.
• x2 y2 2py p2 y2 2py p2
• x2 4py
• The standard form of the equation of a parabola
  that opens upward or downward

F(0,p)
D(x,-p)
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• If pgt0, the parabola opens upward, if plt0 it
  opens downward.
• Parabolas that open to the left or right are
  inverse relations of upward or downward opening
  parabolas.
• So equations of parabolas with vertex (0,0) that
  open to the right or to the left have the
  standard form y2 4px
• If pgt0, the parabola opens to the right and if
  plt0, the parabola opens to the left.

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• p is the focal length of the parabola the
  directed distance from the vertex to the focus of
  the parabola.
• A line segment with endpoints on a parabola is a
  chord of the parabola.
• The value 4p is the focal width of the parabola
  the length of the chord through the focus and
  perpendicular to the axis.

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Parabolas with vertex (0,0)
Standard Equation x24py y2 4px
Opens Up if pgt0 Down if plt0 To right if pgt0 To left if plt0
Focus (0, p) (p, 0)
Directrix y -p x -p
Axis y-axis x-axis
Focal length p p
Focal width 4p 4p
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Example 1a

• Find the focus, the directrix, and focal width of
  the parabola

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Example 1b

• Find the focus, the directrix, and focal width of
  the parabola
• x 2y2

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Example 2

• Find the equation in standard form for a parabola
  whose
• directrix is the line x 5 and focus is the
  point (-5, 0)
• b) directrix is the line y 6 and vertex is (0,0)

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Example 3

• Find the equation in standard form for a parabola
  whose
• vertex is (0,0) and focus is (0, -4)
• b) vertex is (0,0), opens to the left with focal
  width7

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Homework

• Page 641 1,2, 5-20