MTH065 Elementary Algebra II

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MTH065Elementary Algebra II

• Chapter 13
• Conic Sections

Introduction Parabolas (13.1) Circles
(13.1) Ellipses (13.2) Hyperbolas (13.3) Summary
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Where weve been

• MTH 060 Linear Functions Equations
• Single Variable ax b 0
• Solution A single real number.
• Two Variables ax by c
• y mx b
• f(x) mx b
• Solutions Many ordered pairs of real
  numbers.
• Graph A line.

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Where weve been

• MTH 065 Quadratic Functions Equations
• Single Variable ax2 bx c 0
• Solutions 0, 1, or 2 real numbers
• Two Variables y ax2 bx c
• f(x) ax2 bx
  c
• f(x) a(x h)2
  k
• Solutions Many ordered pairs of real
  numbers.
• Graph A parabola.

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Whats missing

• Quadratic Equations that may also include a y2
  term (not all functions).
• Ax2 By2 Cx Dy E 0
• A, B, C, D, E are constants
• A and B not both 0

Note Quadratic equations may also include an xy
term, but the study of such equations requires
trigonometry.
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Parabolas

• y ax2 bx c
• Graphing (complete the square) y a(x - h)2 k
• Vertex (h, k)
• h -b/(2a)
• Orientation
• Open upward a gt 0
• Open downward a lt 0
• Width
• Narrow a gt 1
• Wide a lt 1
• Graphing Vertex One Other Point

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Parabolas

• x ay2 by c
• Graphing (complete the square) x a(y - k)2 h
• Vertex (h, k)
• k -b/(2a)
• Orientation
• Open right a gt 0
• Open left a lt 0
• Width
• Narrow a gt 1
• Wide a lt 1
• Graphing Vertex One Other Point

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Parabolas Special Properties

• Focus
• The point 1/(4a) units from the vertex along the
  axis of symmetry and inside the parabola.
• Reflective property
• Light or any other wave emitted from the focus
  will be reflected in a beam parallel to the axis
  of symmetry.
• A satellite dish, for example, uses this property
  in reverse.

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Ellipses

• Ax2 By2 Cx Dy E 0
• where A B are both positive or both negative.
• Graphing form Complete the squares set equal
  to 1
• Center (h,k)
• 4 Vertices (h a, k), (h, k b)

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Ellipses Special Properties

• Foci
• The two points c units from the center along the
  major axis where c2 a2 b2 if a gt b or c2 b2
  a2 if a lt b.
• Reflective property
• Sound or any other wave emitted from one focus
  will be reflected to the other focus.
• Satellites have elliptical orbits with the object
  being orbited at one of the foci.

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Circles Special Ellipses

• A circle is just an ellipse with a b and a
  single focus at the center (since c2 a2 b2
  0).
• Ax2 Ay2 Cx Dy E 0
• (x h)2 (y k)2 r2
• Center (h, k)
• Radius r

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Hyperbolas

• Ax2 By2 Cx Dy E 0
• where A B have opposite signs.
• Graphing form Complete the squares set equal
  to 1
• Center (h,k)
• 2 Vertices
• 1st form (h a, k)
• 2nd form (h, k b)
• Asymptotes

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Hyperbolas Special Properties

• Foci
• The two points c units from the center inside
  each branch, where c2 a2 b2
• Reflective property
• Light or any other wave emitted from one focus
  towards the other branch will be reflected
  directly away from the other focus (or vice
  versa).
• Hyperbolic mirrors are used in reflector
  telescopes.
• Lampshades cast hyperbolic shadows on a wall.

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Conic Sections SummaryAx2 By2 Cx Dy E
0

• A ? 0 B 0
• Up/Down Parabola
• A 0 B ? 0
• Left/Right Parabola

• A B w/ same sign
• Ellipse
• A B gives a circle
• A B w/ opposite signs
• Hyperbola

To graph complete the squares.
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More Applications of Conics

• Parabolas
• http//www.doe.virginia.gov/Div/Winchester/jhhs/ma
  th/lessons/calc2004/appparab.html
• Ellipses
• http//www.doe.virginia.gov/Div/Winchester/jhhs/ma
  th/lessons/calc2004/appellip.html
• Hyperbolas
• http//www.doe.virginia.gov/Div/Winchester/jhhs/ma
  th/lessons/calc2004/apphyper.html