Holt Algebra 2 10-4 Hyperbolas 10-4 Hyperbolas Holt Algebra 2 Warm Up Warm Up Lesson Presentation...
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Transcript of Holt Algebra 2 10-4 Hyperbolas 10-4 Hyperbolas Holt Algebra 2 Warm Up Warm Up Lesson Presentation...
Holt Algebra 2
10-4 Hyperbolas10-4 Hyperbolas
Holt Algebra 2
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 2
10-4 Hyperbolas
Warm UpMultiply both sides of each equation by the least common multiple to eliminate the denominators.
4x2 – 9y2 = 361.
x2
9– = 1 y2
4
2.
y2
25– = 1 x2
1616y2 – 25x2 = 400
Holt Algebra 2
10-4 Hyperbolas
Write the standard equation for a hyperbola.
Graph a hyperbola, and identify its center, vertices, co-vertices, foci, and asymptotes.
Objectives
Holt Algebra 2
10-4 Hyperbolas
hyperbolafocus of a hyperbolabranch of a hyperbolatransverse axisvertices of a hyperbolaconjugate axisco-vertices of a hyperbola
Vocabulary
Holt Algebra 2
10-4 Hyperbolas
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.
Holt Algebra 2
10-4 Hyperbolas
Find the constant difference for a hyperbola with foci F1 (–8, 0) and F2 (8, 0) and the point on the hyperbola (8, 30).
Example 1: Using the Distance Formula to Find the Constant Difference of a Hyperbola
Definition of the constant difference of a hyperbola.
d = |PF1 – PF2 |
Distance Formula
Substitute.
Simplify.
d = 4 The constant difference is 4.
Holt Algebra 2
10-4 Hyperbolas
Find the constant difference for a hyperbola with foci F1 (0, –10) and F2 (0, 10) and the point on the hyperbola (6, 7.5).
Definition of the constant difference of a hyperbola.
d = |PF1 – PF2 |
Distance Formula
Check It Out! Example 1
Substitute.
Simplify.
d = 12
The constant difference is 12.
Holt Algebra 2
10-4 Hyperbolas
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.
Holt Algebra 2
10-4 Hyperbolas
The standard form of the equation of a hyperbola depends on whether the hyperbola’s transverse axis is horizontal or vertical.
Holt Algebra 2
10-4 Hyperbolas
The values a, b, and c, are related by the equation c2 = a2 + b2. Also note that the length of the transverse axis is 2a and the length of the conjugate is 2b.
Holt Algebra 2
10-4 Hyperbolas
Write an equation in standard form for each hyperbola.
Example 2A: Writing Equations of Hyperbolas
Step 1 Identify the form of the equation.
The graph opens horizontally, so the equation
will be in the form of . x2
a2– = 1 y2
b2
Holt Algebra 2
10-4 Hyperbolas
Example 2A Continued
Step 2 Identify the center and the vertices.
The center of the graph is (0, 0), and the vertices are (–6, 0) and (6, 0), and the co-vertices are (0, –6), and (0, 6). So a = 6, and b = 6.
Step 3 Write the equation.
x2
36– = 1. y2
36
Because a = 6 and b = 6, the equation of the graph is
x2
62– = 1, or y2
62
Holt Algebra 2
10-4 Hyperbolas
Example 2B: Writing Equations of Hyperbolas
The hyperbola with center at the origin, vertex (4, 0), and focus (10, 0).
Step 1 Because the vertex and the focus are on the
horizontal axis, the transverse axis is
horizontal and the equation is in the form
. x2
a2– = 1 y2
b2
Write an equation in standard form for each hyperbola.
Holt Algebra 2
10-4 Hyperbolas
Example 2B Continued
Step 2 Use a = 4 and c = 10; Use c2 = a2 + b2 to solve for b2.
102 = 42 + b2
84 = b2
Substitute 10 for c, and 4 for a.
Step 3 The equation of the hyperbola is . x2
16– = 1 y2
84
Holt Algebra 2
10-4 Hyperbolas
Write an equation in standard form for each hyperbola.
Vertex (0, 9), co-vertex (7, 0)
Step 1 Because the vertex is on the vertical axis, the transverse axis is vertical and the equation is in the form . y2
a2– = 1 x2
b2
Check It Out! Example 2a
Step 2 a = 9 and b = 7.
Step 3 Write the equation.
Because a = 9 and b = 7, the equation of the
graph is , or . y2
92– = 1 x2
72 y2
81– = 1 x2
49
Holt Algebra 2
10-4 Hyperbolas
Vertex (8, 0), focus (10, 0)
Check It Out! Example 2b
x2
a2– = 1 y2
b2
Step 1 Because the vertex and the focus are on
the horizontal axis, the transverse axis is
horizontal and the equation is in the form
.
Write an equation in standard form for each hyperbola.
Holt Algebra 2
10-4 Hyperbolas
Step 2 a = 8 and c = 10; Use c2 = a2 + b2 to solve for b2.
102 = 82 + b2
36 = b2
Substitute 10 for c, and 8 for a.
Step 3 The equation of the hyperbola is . x2
64– = 1 y2
36
Check It Out! Example 2b Continued
Holt Algebra 2
10-4 Hyperbolas
As with circles and ellipses, hyperbolas do not have to be centered at the origin.
Holt Algebra 2
10-4 Hyperbolas
Example 3A: Graphing a Hyperbola
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 horizontal with
center (0, 0).
x2
a2– = 1 y2
b2
x2
49– = 1 y2
9
Holt Algebra 2
10-4 Hyperbolas
Example 3A Continued
Step 2 Because a = 7 and b = 3, the vertices are (–7, 0) and (7, 0) and the co-vertices are (0, –3) and (0, 3).
3 7
Step 3 The equations of the asymptotes are
y = x and y = – x. 3 7
Holt Algebra 2
10-4 Hyperbolas
Example 3A 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.
Holt Algebra 2
10-4 Hyperbolas
Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph.
(x – 3)2
9– = 1(y + 5)2
49
Step 1 The equation is in the form
, so
the transverse axis is horizontal
with center (3, –5).
(x – h)2
a2– = 1(y – k)2
b2
Example 3B: Graphing a Hyperbola
Holt Algebra 2
10-4 Hyperbolas
Example 3B 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) .
Step 3 The equations of the asymptotes are
y + 5 = (x – 3) and y = – (x – 3). 7 3
7 3
Holt Algebra 2
10-4 Hyperbolas
Example 3B 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.
Holt Algebra 2
10-4 Hyperbolas
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 horizontal with center
(0, 0).
x2
a2– = 1 y2
b2
x2
16– = 1 y2
36
Check It Out! Example 3a
Holt Algebra 2
10-4 Hyperbolas
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).
Step 3 The equations of the asymptotes are
y = x and y = – x . 3 2
3 2
Check It Out! Example 3a Continued
Holt Algebra 2
10-4 Hyperbolas
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.
Check It Out! Example 3
Holt Algebra 2
10-4 Hyperbolas
Find the vertices, co-vertices, and asymptotes of each hyperbola, and then graph.
(y + 5)2
1– = 1(x – 1)2
9
Step 1 The equation is in the form
so
the transverse axis is vertical with
center (1, –5).
(y – k)2
a2– = 1(x – h)2
b2
Check It Out! Example 3b
Holt Algebra 2
10-4 Hyperbolas
Step 3 The equations of the asymptotes are
y + 5 = (x – 1) and y + 5 = – (x – 3). 1 3
1 3
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).
Holt Algebra 2
10-4 Hyperbolas
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.
Check It Out! Continued
Holt Algebra 2
10-4 Hyperbolas
Notice that as the parameters change, the graph of the hyperbola is transformed.
Holt Algebra 2
10-4 Hyperbolas
Lesson Quiz: Part I
1. Find the constant difference for a hyperbola with foci (–3.5, 0) and (3.5, 0) and a point on the hyperbola (3.5, 24).
1
2. Write an equation in standard form for a hyperbola with center hyperbola (4, 0), vertex (10, 0), and focus (12, 0).
Holt Algebra 2
10-4 Hyperbolas
Lesson Quiz Part II
3. Find the vertices, co-vertices, and asymptotes of , then graph.
asymptotes:
vertices: (–6, ±5); co-vertices (6, 0), (–18, 0);5
12y = ± (x + 6)
