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. Over Lesson 7-1 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. Over Lesson 7-1 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. Over Lesson 7-1 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. Over Lesson 7-1 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. Over Lesson 7-1 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 7–1) 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 A. Graph the ellipse The equation is in standard form with h = –2, k = 1,

12. Example 1 Graph Ellipses

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

14. Example 1 Graph Ellipses First, write the equation in standard form. 4x 2 + 24x + y 2 – 10y – 3= 0Original equation (4x 2 + 24x) + (y 2 – 10y)= 3Isolate and group like terms. 4(x 2 + 6x) + (y 2 – 10y)= 3Factor. 4(x 2 + 6x + 9) + (y 2 – 10y + 25)= 3 + 4(9) + 25 Complete the squares. 4(x + 3) 2 + (y – 5) 2 = 64Factor and simplify. B. Graph the ellipse 4x 2 + 24x + y 2 – 10y – 3 = 0.

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

16. Example 1 Graph Ellipses

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

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

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. Half the length of major axisHalf the length of minor axis 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. 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 x 2 -term. An equation for the ellipse is. Answer:

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 = 5Simplify.

22. Example 2 Write Equations Given Characteristics Distance formula 2c represents the distance between the foci. c = 3Simplify. Find the value of b. c 2 = a 2 – b 2 Equation relating a, b, and c 3 2 = 5 2 – b 2 a = 5 and c = 3 b= 4Simplify.

23. Example 2 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

24. Example 2 Write Equations Given Characteristics Answer:

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

26. Key Concept 2

27. Example 3 Determine the Eccentricity of an Ellipse Determine the eccentricity of the ellipse given by First, determine the value of c. c 2 = a 2 – b 2 Equation relating a, b, and c c 2 = 64 – 36a 2 = 64 and b 2 = 36 c= Simplify.

28. Example 3 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

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= cMultiply.

31. Example 4 Use Eccentricity Use the values of c and a to determine b. c 2 = a 2 – b 2 Equation relating a, b, and c. 9.0146 2 = 19.18 2 – b 2 c = 9.0146, a = 19.18 33.86≈ bMultiply. 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 9x 2 + 4y 2 + 8y – 32= 0Original equation 9x 2 + 4(y 2 + 2y)= 32Isolate like terms. 9x 2 + 4(y 2 + 2y + 1)= 32 + 4(1)Complete the square. 9x 2 + 4(y + 1) 2 = 36Factor 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.

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

36. Example 5 B. Write x 2 + 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= 0Original equation x 2 + 4x + 4 – 4y + 16= 0 + 4Complete the square. (x + 2) 2 – 4y + 16= 4Factor and simplify. (x + 2) 2 = 4y – 12Add 4y – 16 to each side. (x + 2) 2 = 4(y – 3)Factor. Because only one term is squared, the conic selection is a parabola.

37. Example 5 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= 0Original equation x 2 + 2x + y 2 – 6y= 6Isolate like terms. x 2 + 2x + 1 + y 2 – 6y + 9= 6 + 1 + 9Complete the square. (x + 1) 2 + (y – 3) 2 = 16Factor and simplify. Because the equation is of the form (x – h) 2 + (y – k) 2 = r 2, the conic selection is a circle.

38. Example 5 Write 16x 2 + y 2 + 4y – 60 = 0 in standard form. Identify the related conic. A. B.16x 2 + (y + 2) 2 = 64; circle C. D.16x 2 + (y + 2) 2 = 64; ellipse

39. End of the Lesson