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

Over Lesson 7-1

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

Over Lesson 7-1

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

Over Lesson 7-1

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)

Over Lesson 7-1

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)

Over Lesson 7-1

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)

You analyzed and graphed parabolas. (Lesson 7–1)

• Analyze and graph equations of ellipses and circles.

• Use equations to identify ellipses and circles.

• ellipse

• foci

• major axis

• center

• minor axis

• vertices

• co-vertices

• eccentricity

Graph Ellipses

A. Graph the ellipse

The equation is in standard form with h = –2, k = 1,

Graph Ellipses

Graph Ellipses

Answer:

Graph the center, vertices, and axes. Then make a table of values to sketch the ellipse.

xxx-new art (graph)

Graph Ellipses

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) + 25Complete thesquares.

4(x + 3)2 + (y – 5)2 = 64 Factor and simplify.

B. Graph the ellipse 4x 2 + 24x + y

2 – 10y – 3 = 0.

Graph Ellipses

Divide each side by 64.

Graph Ellipses

Graph Ellipses

Answer:

Graph the center, vertices, foci, and axes. Then make a table of values to sketch the ellipse.

Graph the ellipse 144x

2 + 1152x + 25y 2 – 300y – 396 = 0.

A. C.

B. D.

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.

Half the length of major axis Half the length of minor axis

The center of the ellipse is at the midpoint of the major axis.

Write Equations Given Characteristics

Midpoint formula

= (2, –2) Simplify.

The y-coordinates are the same for both endpoints of

the major axis, so the major axis is horizontal and the

value of a belongs with the x2-term. An equation for

the ellipse is .

Answer:

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.

Write Equations Given Characteristics

Distance formula

2c represents the distance between the foci.

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.

Write Equations Given Characteristics

Midpoint formula

The vertices are equidistant from the center.

= (3, 1) Simplify.

The x-coordinates are the same for both endpoints of

the major axis, so the major axis is vertical and the

value of a belongs with the y 2-term. An equation for

the ellipse is

Write Equations Given Characteristics

Answer:

Write an equation for an ellipse with co-vertices (–8, 6) and (4, 6) and major axis of length 18.

A.

B.

C.

D.

Determine the Eccentricity of an Ellipse

Determine the eccentricity of the ellipse given by

First, determine the value of c.

c2 = a2 – b2 Equation relating a, b,and c 

c2 = 64 – 36 a2 = 64 and b2 = 36

c = Simplify.

Determine the Eccentricity of an Ellipse

Answer: about 0.66

Use the values of c and a to find the eccentricity.

Eccentricity equation

The eccentricity of the ellipse is about 0.66.

a = 8

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

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.

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

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

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.

A. Write 9x 2 + 4y

2 + 8y – 32 = 0 in standard form. Identify the related conic.

Determine Types of Conics

Divide each side by 36.

Determine Types of Conics

Answer:

the conic selection is an ellipse.

B. Write x2 + 4x – 4y + 16 = 0 in standard form. Identify the related conic selection.

Answer: (x + 2)2 = 4(y – 3); parabola

Determine Types of Conics

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 eachside.

(x + 2)2 = 4(y – 3) Factor.

Because only one term is squared, the conic selection is a parabola.

C. Write x 2 + y

2 + 2x – 6y – 6 = 0 in standard form. Identify the related conic.

Answer: (x + 1)2 + (y – 3)2 = 16; circle

Determine Types of Conics

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 thesquare.

(x + 1)2 + (y – 3)2 = 16 Factor andsimplify.

Because the equation is of the form (x – h)2 + (y – k)2 = r

2, the conic selection is a circle.

Write 16x 2 + y

2 + 4y – 60 = 0 in standard form. Identify the related conic.

A.

B. 16x2 + (y + 2)2 = 64; circle

C.

D. 16x2 + (y + 2)2 = 64; ellipse