1. Splash Screen

2. 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

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. Graph x 2 + y 2 + 2x – 6y – 6 = 0 in standard form.
x 2 + y 2 + 2x – 6y – 6 = 0 Original equation
x 2 + 2x + y 2 – 6y = 6 Group like terms.
x 2 + 2x y 2 – 6y + 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. Vertices (h ±c, k)  (1, 1) and (-5, 1)
Graph Ellipses
Graph the ellipse
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 the ellipse 4x 2 + 24x + y 2 – 10y – 3 = 0.
Graph Ellipses
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 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.
THIS IS WHAT MAKES ELLIPSE INSTEAD OF CIRCLE!!!
Example 1

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

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

14. x 2 + 4x – 4y + 16 = 0 Original equation
Determine Types of Conics
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

15. 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

16. 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).
Length of major axis = 2a
length of minor axis = 2b
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.
Answer:
Example 2

17. a = 5  since vertices are on x = 3, vertical (h, k ± a)
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. 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

19. Key Concept 2

20. 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 using c2= a2 – b2
c =
Eccentricity equation
The eccentricity of the ellipse is about 0.66.
Example 3

21. Use c2 = a2 – b2 to find the value of b. 33.86 ≈ b Minor Axis ≈ 67.72
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.
= 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
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. Homework
Day 1: P. 438 #5, 27-29, 45, 66, 94
Day 2: P. 438 # 7, 11, 17, 39, 55, 65