1. Splash Screen

2. 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
Lesson Menu

3. Write y 2 – 6y – 4x + 17 = 0 in standard form
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
5-Minute Check 1

4. Write y 2 – 6y – 4x + 17 = 0 in standard form
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
5-Minute Check 1

5. Write x 2 + 8x – 4y + 8 = 0 in standard form
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-Minute Check 2

6. Write x 2 + 8x – 4y + 8 = 0 in standard form
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-Minute Check 2

7. 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)
5-Minute Check 3

8. 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)
5-Minute Check 3

9. 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)
5-Minute Check 4

10. 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)
5-Minute Check 4

11. 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)
5-Minute Check 5

12. 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)
5-Minute Check 5

13. You analyzed and graphed parabolas. (Lesson 7–1)
Analyze and graph equations of ellipses and circles.
Use equations to identify ellipses and circles.
Then/Now

14. ellipse foci major axis center minor axis vertices co-vertices
eccentricity
Vocabulary

15. Key Concept 1

16. The equation is in standard form with h = –2, k = 1,
Graph Ellipses
A. Graph the ellipse
The equation is in standard form with h = –2, k = 1,
Example 1

17. Graph Ellipses
Example 1

18. Graph Ellipses
Graph the center, vertices, and axes. Then make a table of values to sketch the ellipse.
Answer:
Example 1

19. Graph Ellipses
Graph the center, vertices, and axes. Then make a table of values to sketch the ellipse.
Answer:
xxx-new art (graph)
Example 1

20. B. Graph the ellipse 4x 2 + 24x + y 2 – 10y – 3 = 0.
Graph Ellipses
B. Graph the ellipse 4x x + y 2 – 10y – 3 = 0.
First, write the equation in standard form.
4x x + 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) Complete the squares.
4(x + 3)2 + (y – 5)2 = 64 Factor and simplify.
Example 1

21. Graph Ellipses
Divide each side by 64.
Example 1

22. Graph Ellipses
Example 1

23. Graph Ellipses
Graph the center, vertices, foci, and axes. Then make a table of values to sketch the ellipse.
Answer:
Example 1

24. Graph Ellipses
Graph the center, vertices, foci, and axes. Then make a table of values to sketch the ellipse.
Answer:
Example 1

25. Graph the ellipse 144x 2 + 1152x + 25y 2 – 300y – 396 = 0.C. B. D.
Example 1

26. Graph the ellipse 144x 2 + 1152x + 25y 2 – 300y – 396 = 0.C. B. D.
Example 1

27. Use the major and minor axes to determine a and b.
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.
Example 2

28. Midpoint formula = (2, –2) Simplify.
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:
Example 2

29. Midpoint formula = (2, –2) Simplify.
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:
Example 2

30. 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.
Example 2

31. 2c represents the distance between the foci.
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.
Example 2

32. The vertices are equidistant from the center.
Write Equations Given Characteristics
The vertices are equidistant from the center.
Midpoint formula
= (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
Example 2

33. Write Equations Given Characteristics
Answer:
Example 2

34. Write Equations Given Characteristics
Answer:
Example 2

35. Write an equation for an ellipse with co-vertices (–8, 6) and (4, 6) and major axis of length 18.B. C. D.
Example 2

36. Write an equation for an ellipse with co-vertices (–8, 6) and (4, 6) and major axis of length 18.B. C. D.
Example 2

37. Key Concept 2

38. Determine the eccentricity of the ellipse given by
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.
Example 3

39. Use the values of c and a to find the eccentricity.
Determine the Eccentricity of an Ellipse
Use the values of c and a to find the eccentricity.
Eccentricity equation
a = 8
The eccentricity of the ellipse is about 0.66.
Answer:
Example 3

40. Use the values of c and a to find the eccentricity.
Determine the Eccentricity of an Ellipse
Use the values of c and a to find the eccentricity.
Eccentricity equation
a = 8
The eccentricity of the ellipse is about 0.66.
Answer: about 0.66
Example 3

41. 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
Example 3

42. 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
Example 3

43. Definition of eccentricity
Use Eccentricity
ASTRONOMY The eccentricity of the orbit of Uranus is Its orbit around the Sun has a major axis length of AU (astronomical units). What is the length of the minor axis of the orbit?
The major axis is 38.36, so a = Use the eccentricity to find the value of c.
Definition of eccentricity
e = 0.47, a = 19.18
= c Multiply.
Example 4

44. Use the values of c and a to determine b.
Use Eccentricity
Use the values of c and a to determine b.
c2 = a2 – b2 Equation relating a, b, and c.
= – b2 c = , a = 19.18
33.86 ≈ b Multiply.
Answer:
Example 4

45. Use the values of c and a to determine b.
Use Eccentricity
Use the values of c and a to determine b.
c2 = a2 – b2 Equation relating a, b, and c.
= – b2 c = , a = 19.18
33.86 ≈ b Multiply.
Answer:
Example 4

46. PARKS A lake in a park is elliptically-shaped
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
Example 4

47. PARKS A lake in a park is elliptically-shaped
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
Example 4

48. Key Concept 3

49. 9x 2 + 4y 2 + 8y – 32 = 0 Original equation
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) = (1) Complete the square.
9x 2 + 4(y + 1)2 = 36 Factor and simplify.
Divide each side by 36.
Example 5

50. the conic selection is an ellipse.
Determine Types of Conics
the conic selection is an ellipse.
Answer:
Example 5

51. the conic selection is an ellipse.
Determine Types of Conics
the conic selection is an ellipse.
Answer:
Example 5

52. x 2 + 4x – 4y + 16 = 0 Original equation
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 = 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:
Example 5

53. x 2 + 4x – 4y + 16 = 0 Original equation
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 = 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
Example 5

54. x 2 + y 2 + 2x – 6y – 6 = 0 Original equation
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 y 2 – 6y + 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:
Example 5

55. x 2 + y 2 + 2x – 6y – 6 = 0 Original equation
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 y 2 – 6y + 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
Example 5

56. Write 16x 2 + y 2 + 4y – 60 = 0 in standard form
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
Example 5

57. Write 16x 2 + y 2 + 4y – 60 = 0 in standard form
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
Example 5

58. End of the Lesson