Today in Precalculus

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Today in Precalculus

• Turn in graded worksheet
• Notes Conic Sections - Hyperbolas
• Homework

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Hyperbolas

• Definition A hyperbola is the set of all points
  in a plane whose distances from two fixed points
  in a plane have a constant difference. The fixed
  points are the foci (F). The line through the
  foci is the focal axis. The point on the focal
  axis midway between the two foci is the center
  (C). The points on the hyperbola that intersect
  with the focal axis are the vertices (V).

F
F
V
V
C
focal axis
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• Standard form for the equation of a hyperbola
  centered at the origin with the x-axis as its
  focal axis is
• There is a pythagorean relationship between a,b,
  and c
• c2 a2 b2

F(-c,0)
F (c,0)
(-a,0)
(a,0)
C(0,0)
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• A line segment with endpoints on an hyperbola is
  a chord of the hyperbola
• The chord lying on the focal axis connecting the
  vertices is the transverse axis of the hyperbola
  and has a length of 2a.
• The value for a is the semitransverse axis.
• The segment through the center perpendicular to
  the focal axis is the conjugate axis of the
  hyperbola and has a length of 2b.
• The value of b is the semiconjugate axis
• The hyperbola also has two slant
• asymptotes whose equations
• depend on a and b.

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• A hyperbola centered at the origin with the
  y-axis as its focal axis has the form

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Hyperbola with center (0,0)
Standard Equation
Focal axis x-axis y-axis
Foci (c, 0) (0, c)
Vertices (a, 0) (0, a)
Semitransverse axis a a
Semiconjugate axis b b
Pythagorean relation c2 a2 b2 c2 a2 b2
Asymptotes
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Example 1

• Find the vertices and foci of the hyperbola 4x2
  9y2 36
• Vertices (-3, 0), (3, 0)
• c2 9 4 13
• c 3.6
• Foci (-3.6, 0), (3.6,0)

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Example 2

• Find an equation of the hyperbola with foci (0,
  ) and transverse axis length
• From the foci c and focal axis is y-axis
• 2a
• a
• 17 13 b2
• 4 b2

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Sketching hyperbolas

• Vertices (0, -3.6), (0, 3.6) Points (-2, 0), (2,0)

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Graphing a hyperbola

• Like the other conic sections, must solve the
  equation for y

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Homework

• Pg 663 1, 2, 5, 6, 11-14, 23-26