Reminder: Multiplying both sides of each equation by the least common multiple eliminates the fractions.

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Reminder Multiplying both sides of each equation
by the least common multiple eliminates the
fractions.
4x2 9y2 36
16y2 25x2 400
Objectives
Write the standard equation for a
hyperbola. Graph a hyperbola, and identify its
center, vertices, co-vertices, foci, and
asymptotes.
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Vocabulary
hyperbola branch and focus of a
hyperbola transverse axis conjugate
axis vertices and co-vertices of a hyperbola
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Notes
1. Find the vertices, co-vertices, and asymptotes
of , then graph.
2. Write an equation in standard form for a
hyperbola with center hyperbola (4, 0), vertex
(10, 0), and focus (12, 0).
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What would happen if you pulled the two foci of
an ellipse so far apart that they moved outside
the ellipse? The result would be a hyperbola,
another conic section.
A hyperbola is a set of points P(x, y) in a plane
such that the difference of the distances from P
to fixed points F1 and F2, the foci, is constant.
For a hyperbola, d PF1 PF2 , where d is the
constant difference. You can use the distance
formula to find the equation of a hyperbola.
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The standard form of the equation of a hyperbola
depends on whether the hyperbolas transverse
axis is horizontal or vertical.
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As the graphs in the following table show, a
hyperbola contains two symmetrical parts called
branches. A hyperbola also has two axes of
symmetry. The transverse axis of symmetry
contains the vertices and, if it were extended,
the foci of the hyperbola. The vertices of a
hyperbola are the endpoints of the transverse
axis. The conjugate axis of symmetry separates
the two branches of the hyperbola. The
co-vertices of a hyperbola are the endpoints of
the conjugate axis. The transverse axis is not
always longer than the conjugate axis.
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The values a, b, and c, are related by the
equation c2 a2 b2. Also note that the
length of the trans-verse axis is 2a and the
length of the conjugate is 2b.
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Example 1A Graphing a Hyperbola
Find the vertices, co-vertices, and asymptotes of
each hyperbola, and then graph
Step 1 Because a 7 and b 3, the vertices are
(7, 0) and (7, 0) and the co-vertices are
(0, 3) and (0, 3).
Step 2 The equations of the asymptotes are and
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Example 2A Continued
Step 3 Draw a box by using the vertices and
co-vertices. Draw the asymptotes through the
corners of the box.
Step 4 Draw the hyperbola by using the vertices
and the asymptotes.
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As with circles and ellipses, hyperbolas do not
have to be centered at the origin.
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Example 1B Graphing a Hyperbola
Find the vertices, co-vertices, and asymptotes of
each hyperbola, and then graph.
Step 1 Identify transverse axis as horizontal
with center (3, 5).
Step 2 Because a 3 and b 7, the vertices are
(0, 5) and (6, 5) and the co-vertices are (3,
12) and (3, 2).
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Example 1B Continued
Step 2 Because a 3 and b 7, the vertices are
(0, 5) and (6, 5) and the co-vertices are (3,
12) and (3, 2) .
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Example 1B Continued
Step 4 Draw a box by using the vertices and
co-vertices. Draw the asymptotes through the
corners of the box.
Step 5 Draw the hyperbola by using the vertices
and the asymptotes.
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Example 2A Writing Equations of Hyperbolas
Write an equation in standard form for each
hyperbola.
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Example 2B Writing Equations of Hyperbolas
Write an equation for the hyperbola with center
at the origin, vertex (4, 0), and focus (10, 0).
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Example 2B Continued
Step 2 Use a 4 and c 10 Use c2 a2 b2 to
solve for b2.
102 42 b2
Substitute 10 for c, and 4 for a.
84 b2
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Notes
1. Find the vertices, co-vertices, and asymptotes
of , then graph.
2. Write an equation in standard form for a
hyperbola with center hyperbola (4, 0), vertex
(10, 0), and focus (12, 0).
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Hyperbolas Extra Info
The following power-point slides contain extra
examples and information.
Reminder Lesson Objectives Write the standard
equation for a hyperbola. Graph a hyperbola, and
identify its center, vertices, co-vertices, foci,
and asymptotes.
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Notice that as the parameters change, the graph
of the hyperbola is transformed.
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Check It Out! Example 3a Graphing
Find the vertices, co-vertices, and asymptotes of
each hyperbola, and then graph.
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Check It Out! Example 3a Continued
Step 2 Because a 4 and b 6, the vertices are
(4, 0) and (4, 0) and the co-vertices are
(0, 6) and . (0, 6).
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Check It Out! Example 3
Step 4 Draw a box by using the vertices and
co-vertices. Draw the asymptotes through the
corners of the box.
Step 5 Draw the hyperbola by using the vertices
and the asymptotes.
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Check It Out! Example 3b Graphing
Find the vertices, co-vertices, and asymptotes of
each hyperbola, and then graph.
Step 1 The equation is in the form so the
transverse axis is vertical with center (1, 5).
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Check It Out! Example 3b Continued
Step 2 Because a 1 and b 3, the vertices are
(1, 4) and (1, 6) and the co-vertices are (4,
5) and (2, 5).
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Check It Out! Continued
Step 4 Draw a box by using the vertices and
co-vertices. Draw the asymptotes through the
corners of the box.
Step 5 Draw the hyperbola by using the vertices
and the asymptotes.
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Check It Out! Example 2a Writing Equations
Write an equation in standard form for each
hyperbola.
Vertex (0, 9), co-vertex (7, 0)
Step 2 a 9 and b 7.
Step 3 Write the equation.
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Check It Out! Example 2b Writing Equations
Write an equation in standard form for each
hyperbola.
Vertex (8, 0), focus (10, 0)
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Check It Out! Example 2b Continued
Step 2 a 8 and c 10 Use c2 a2 b2 to
solve for b2.
102 82 b2
Substitute 10 for c, and 8 for a.
36 b2