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Splash Screen

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Title: Glencoe PreCalculus Last modified by: architect Created Date: 12/6/2007 2:31:51 PM Document presentation format: On-screen Show Company: Madeira Station LLC – PowerPoint PPT presentation

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Title: Splash Screen


1
Splash Screen
2
Lesson Menu
Five-Minute Check (over Lesson 7-1) Then/Now New
Vocabulary Key Concept Standard Forms of
Equations for Ellipses Example 1 Graph
Ellipses Example 2 Write Equations Given
Characteristics Key Concept Eccentricity Example
3 Determine the Eccentricity of an
Ellipse Example 4 Real World Example Use
Eccentricity Key Concept Standard Form of
Equations for Circles Example 5 Determine Types
of Conics
3
5-Minute Check 1
Write y 2 6y 4x 17 0 in standard form.
Identify the vertex, focus, axis of symmetry, and
directrix.
A. (y 3)2 4(x 2) vertex (2, 3) focus
(3, 3) axis of symmetry y 3 directrix x 1
B. (y 3)2 4(x 2) vertex (2, 3) focus
(1, 3) axis of symmetry y 3 directrix x
3 C. (y 3)2 4(x 2) vertex (3, 2) focus
(3, 3) axis of symmetry y 3 directrix x 1
D. (y 3)2 4(x 2) vertex (2, 3) focus
(6, 3) axis of symmetry y 3 directrix x
2
4
5-Minute Check 2
Write x 2 8x 4y 8 0 in standard form.
Identify the vertex, focus, axis of symmetry, and
directrix.
A. (x 4)2 4(y 2) vertex (4, 2) focus
(3, 2) axis of symmetry x 2 directrix y
5 B. (x 4)2 4(y 2) vertex (4, 2)
focus (4, 1) axis of symmetry x 4
directrix y 3 C. (x 4)2 4(y 2)
vertex (4, 2) focus (4, 3) axis of
symmetry x 4 directrix y 1 D. (x 4)2
4(y 2) vertex (4, 2) focus (4, 2)
axis of symmetry x 4 directrix y 6
5
5-Minute Check 3
Write an equation for a parabola with focus F (2,
5) and vertex V (2, 3).
A. (x 2)2 8(y 5) B. (x 2)2 8(y 3)
C. (x 2)2 2(y 3) D. (x 2)2 8(y 3)
6
5-Minute Check 4
Write an equation for a parabola with focus F (2,
2) and vertex V (1, 2).
A. (x 2)2 12(y 2) B. (y 2)2 12(x 2)
C. (y 2)2 12(x 1) D. (x 1)2 12(y 2)
7
5-Minute Check 5
Which of the following equations represents a
parabola with focus (3, 7) and vertex (-3, 2)?
A. (x 3)2 5(y 2) B. (y 3)2 5(x 2)
C. (x 3)2 20(y 2) D. (y 2)2 20(x 3)
8
Then/Now
You analyzed and graphed parabolas. (Lesson 71)
  • Analyze and graph equations of ellipses and
    circles.
  • Use equations to identify ellipses and circles.

9
Vocabulary
  • ellipse
  • foci
  • major axis
  • center
  • minor axis
  • vertices
  • co-vertices
  • eccentricity

10
Key Concept 1
11
Example 1
Graph Ellipses
12
Example 1
Graph Ellipses
13
Example 1
Graph Ellipses
Graph the center, vertices, and axes. Then make a
table of values to sketch the ellipse.
Answer
xxx-new art (graph)
14
Example 1
Graph Ellipses
B. Graph the ellipse 4x 2 24x y 2 10y 3
0.
First, write the equation in standard form.
4x 2 24x y 2 10y 3 0 Original
equation (4x2 24x) (y 2 10y) 3 Isolate
and group like terms. 4(x 2 6x) (y 2
10y) 3 Factor. 4(x 2 6x 9) (y 2 10y
25) 3 4(9) 25 Complete
the squares. 4(x 3)2 (y 5)2 64 Factor
and simplify.
15
Example 1
Graph Ellipses
Divide each side by 64.
16
Example 1
Graph Ellipses
17
Example 1
Graph Ellipses
Graph the center, vertices, foci, and axes. Then
make a table of values to sketch the ellipse.
Answer
18
Example 1
Graph the ellipse 144x 2 1152x 25y 2 300y
396 0.
19
Example 2
Write Equations Given Characteristics
A. Write an equation for an ellipse with a major
axis from (5, 2) to (1, 2) and a minor axis
from (2, 0) to (2, 4).
Use the major and minor axes to determine a and b.
The center of the ellipse is at the midpoint of
the major axis.
20
Example 2
Write Equations Given Characteristics
Midpoint formula
(2, 2) Simplify.
21
Example 2
Write Equations Given Characteristics
B. Write an equation for an ellipse with vertices
at (3, 4) and (3, 6) and foci at (3, 4) and (3,
2)
The length of the major axis, 2a, is the distance
between the vertices.
Distance formula
a 5 Simplify.
22
Example 2
Write Equations Given Characteristics
2c represents the distance between the foci.
Distance formula
c 3 Simplify. Find the value of b. c2 a2
b2 Equation relating a, b, and c 32 52
b2 a 5 and c 3 b 4 Simplify.
23
Example 2
Write Equations Given Characteristics
The vertices are equidistant from the center.
Midpoint formula
(3, 1) Simplify.
24
Example 2
Write Equations Given Characteristics
25
Example 2
Write an equation for an ellipse with co-vertices
(8, 6) and (4, 6) and major axis of length 18.
26
Key Concept 2
27
Example 3
Determine the Eccentricity of an Ellipse
First, determine the value of c. c2 a2
b2 Equation relating a, b, and c  c2 64
36 a2 64 and b2 36
28
Example 3
Determine the Eccentricity of an Ellipse
Use the values of c and a to find the
eccentricity.
Eccentricity equation
The eccentricity of the ellipse is about 0.66.
Answer about 0.66
29
Example 3
Determine the eccentricity of the ellipse given
by 36x 2 144x 49y 2 98y 1571.
A. 0.27 B. 0.36 C. 0.52 D. 0.60
30
Example 4
Use Eccentricity
ASTRONOMY The eccentricity of the orbit of
Uranus is 0.47. Its orbit around the Sun has a
major axis length of 38.36 AU (astronomical
units). What is the length of the minor axis of
the orbit?
The major axis is 38.36, so a 19.18. Use the
eccentricity to find the value of c.
Definition of eccentricity
e 0.47, a 19.18
9.0146 c Multiply.
31
Example 4
Use Eccentricity
Use the values of c and a to determine b.
c2 a2 b2 Equation relating a, b, and
c. 9.01462 19.182 b2 c 9.0146, a
19.18 33.86 b Multiply.
Answer 33.86
32
Example 4
PARKS A lake in a park is elliptically-shaped.
If the length of the lake is 2500 meters and the
width is 1500 meters, find the eccentricity of
the lake.
A. 0.2 B. 0.4 C. 0.6 D. 0.8
33
Key Concept 3
34
Example 5
Determine Types of Conics
A. Write 9x 2 4y 2 8y 32 0 in standard
form. Identify the related conic.
9x 2 4y 2 8y 32 0 Original equation 9x2
4(y 2 2y) 32 Isolate like terms. 9x2 4(y
2 2y 1) 32 4(1) Complete the square. 9x
2 4(y 1)2 36 Factor and simplify.
Divide each side by 36.
35
Example 5
Determine Types of Conics
36
Example 5
Determine Types of Conics
B. Write x2 4x 4y 16 0 in standard form.
Identify the related conic selection.
x 2 4x 4y 16 0 Original equation x 2
4x 4 4y 16 0 4 Complete the square. (x
2)2 4y 16 4 Factor and simplify. (x
2)2 4y 12 Add 4y 16 to each side. (x
2)2 4(y 3) Factor. Because only one term is
squared, the conic selection is a parabola.
Answer (x 2)2 4(y 3) parabola
37
Example 5
Determine Types of Conics
C. Write x 2 y 2 2x 6y 6 0 in standard
form. Identify the related conic.
x 2 y 2 2x 6y 6 0 Original equation x
2 2x y 2 6y 6 Isolate like terms. x 2
2x 1 y 2 6y 9 6 1 9 Complete
the square. (x 1)2 (y 3)2 16 Factor
and simplify. Because the equation is of the
form (x h)2 (y k)2 r 2, the conic
selection is a circle.
Answer (x 1)2 (y 3)2 16 circle
38
Example 5
Write 16x 2 y 2 4y 60 0 in standard form.
Identify the related conic.
39
End of the Lesson
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