11'4 Hyperbolas

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11.4 Hyperbolas

• Hyperbolas are similar to ellipses except, the
  difference of the distances between a point on a
  hyperbola and the two fixed points is constant.
• Hyperbolas can have a horizontal transverse axis,
  where the hyperbola is on the x-axis.
• Or a vertical transverse axis, where the
  hyperbola is on the y-axis.
• a is always on the transverse axis, which is
  similar to the major axis in an ellipse.
• In this section, all hyperbolas will have a
  center of (0,0).
• Hyperbolas have asymptotes, hidden lines that
  guide the graph, and foci.
• The asymptotes are constructed using a hidden box
  whose sides are the values a and b.

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Horizontal Transverse Axis
Center (0,0)
y
(0,b)
(-a,0)
(a,0)
x
(c,0)
(-c,0)
(0,-b)

• The box representing the asymptotes should be
  drawn dotted.
• The asymptotes are drawn very lightly through the
  diagonals of the box.
• The hyperbola is then drawn through points (-a,0)
  and (a,0), using the asymptotes as guides.

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Vertical Transverse Axis
(0,a)
(c,0)
Center (0,0)
y
(-b,0)
(b,0)
x
(-c,0)
(0,-a)

• The box representing the asymptotes should be
  drawn dotted.
• The asymptotes are drawn very lightly through the
  diagonals of the box.
• The hyperbola is then drawn through points (-a,0)
  and (a,0), using the asymptotes as guides.

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Example 1

• Write the equation for the hyperbola given the
  following information foci (-3,0) (3,0)
  vertices (-2,0) (2,0)
• Solution
• From the information given, it can be seen that
  the foci is on the
• x-axis therefore the hyperbola has a horizontal
  transverse axis.

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Example 2 Hyperbola Sketching

• Sketch the hyperbola given 4x2 y2 16
• Solution Put the equation in standard form.
• Find a, b, and c.
• Center (0,0)

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Sketching the Hyperbola

• Plot the points a, b, and the foci.
• Draw a dotted box through these points forming a
  rectangle.
• Lightly draw diagonal lines through the box for
  the asymptotes.
• Using the asymptotes as a guide, sketch the two
  sides of the hyperbola through point a.
• Label the graph with the equation for the
  hyperbola.

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Example 3

• Find the foci of the hyperbola given the
  information
• vertices (0,-3) (0,3) asymptotes y 2x, y
  -2x
• Solution Vertices (0,y) shows vertical
  transverse axis.