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Splash Screen. Write y 2 – 6 y – 4 x + 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

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  1. Splash Screen

  2. Write y2– 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

  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) 5-Minute Check 3

  4. Which of the following equations represents a parabola with focus (–3, 7) and directrix y = -3? 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

  5. 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

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

  7. Key Concept 3

  8. Graphx2 + y2 + 2x – 6y – 6 = 0 in standard form. x2 + y2 + 2x– 6y– 6 = 0 Original equation x2 + 2x+ y2– 6y= 6 Group like terms. x2 + 2x + 1 + y2– 6y + 9 = 6 + 1+ 9 Complete the square. (x + 1)2 + (y– 3)2 = 16 Factor and simplify. Answer:Circle with center (-1, 3) and radius 4. Example 5

  9. Key Concept 1

  10. Graph the ellipse Graph Ellipses Center (-2, 1) Vertices (h ±c, k)  (1, 1) and (-5, 1) Major Axis (y = k), length 2a y = 1, length 6 Minor Axis (x = h), length 2b  x = -2, length 4 Foci (h ±c, k)  (-2 ±√5, 1) Example 1

  11. Graph Ellipses Graph the ellipse 4x2 + 24x + y2 – 10y – 3 = 0. First, write the equation in standard form. 4x2 +24x +y2 –10y –3 = 0 Original equation (4x2+ 24x) + (y2– 10y) = 3 Group like terms. 4(x2 + 6x) + (y2– 10y) = 3 Factor. 4(x2 + 6x + 9) + (y2– 10y + 25) = 3 + 4(9) + 25 Complete the squares. 4(x + 3)2 + (y– 5)2 = 64 Factor and simplify. THIS IS WHAT MAKES ELLIPSE INSTEAD OF CIRCLE!!! Example 1

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

  13. A. C. B. D. Graph the ellipse 144x2 + 1152x + 25y2 – 300y – 396 = 0. Example 1

  14. Determine Types of Conics Write x2 + 4x – 4y + 16 = 0 in standard form. Identify the related conic selection. x2 + 4x– 4y + 16 = 0 Original equation x2 + 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 Example 5

  15. A. B. 16x2 + (y + 2)2 = 64; circle C. D. 16x2 + (y + 2)2 = 64; ellipse Write 16x2 + y2 + 4y – 60 = 0 in standard form. Identify the related conic. Example 5

  16. Length of major axis = 2a length of minor axis = 2b Answer: Write Equations Given Characteristics 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). Major has constant y. Major Axis y = -2, Horizontal Ellipse. Center will be middle of each axis, use Midpoint Formula and plug into Horizontal Equation. Example 2

  17. Write Equations Given Characteristics 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 is the distance between vertices, which is = 2a. a = 5  since vertices are on x = 3, vertical(h, k± a) h = 3 and k = 1 The length between foci = 2c. c = 3 Find the value of b using c2 = a2 – b2 b = 4 Example 2

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

  19. Key Concept 2

  20. Determine the eccentricity of the ellipse given by c = Determine the Eccentricity of an Ellipse First, determine the value of c using c2= a2–b2 Eccentricity equation The eccentricity of the ellipse is about 0.66. Example 3

  21. Use Eccentricity ASTRONOMYThe 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. 9.0146 = c Use c2= a2–b2 to find the value of b. 33.86 ≈ b Minor Axis ≈ 67.72 Example 4

  22. 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

  23. Day 1: P. 438 #5, 27-29, 45, 66, 94 Day 2: P. 438 # 7, 11, 17, 39, 55, 65 Homework

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