1. ELLIPSE – a conic section formed by the intersection of a right circular cone and a plane.

2. ELLIPSE Center is at ( 0, 0 )

3. ELLIPSE Center is at ( h, k )

4. ELLIPSE Center is at ( h, k ) Center is at ( 0, 0 ) Standard Form :

5. ELLIPSE - graphs When a 2 > b 2 -x +x +y -y

6. ELLIPSE - graphs When a 2 > b 2 Major axis -x +x +y -y

7. ELLIPSE - graphs When a 2 > b 2 Major axis -x +x +y -y Minor axis

8. ELLIPSE - graphs When a 2 > b 2 Major axis -x +x +y -y Minor axis Major axis vertices

9. ELLIPSE - graphs When a 2 > b 2 Major axis -x +x +y -y Minor axis Major axis verticesMinor axis vertices

10. ELLIPSE - graphs When a 2 > b 2 -x +x +y -y Foci - fixed coordinate points inside the ellipse - used to create the ellipse - the distance from one of the foci, to ANY point on the ellipse, to the other foci is equal - to find the foci Foci

11. ELLIPSE - graphs When a 2 > b 2 -x +x +y -y Foci - fixed coordinate points inside the ellipse - used to create the ellipse - the distance from one of the foci, to ANY point on the ellipse, to the other foci is equal - the green distance = the black distance

12. ELLIPSE - graphs When b 2 > a 2 -x +x +y -y

13. ELLIPSE - graphs When b 2 > a 2 -x +x +y -y Major axis

14. ELLIPSE - graphs When b 2 > a 2 -x +x +y -y Major axis Minor axis

15. ELLIPSE - graphs When b 2 > a 2 -x +x +y -y Minor axis vertices Major axis vertices

16. ELLIPSE - graphs When b 2 > a 2 -x +x +y -y Foci

17. When working with ellipses, we will always find the following : Center ( h, k ) a = √a 2 Major Axis vertices ( x, y ), ( x, y ) b = √b 2 Minor Axis vertices ( x, y ), ( x, y ) c = Foci vertices ( x, y ) “h” is ALWAYS adjusted by “a” “k” is ALWAYS adjusted by “b” The Foci ALWAYS lies on the major axis NOTE : I’d write these parameters down somewhere, the test problems are EXACTLY like these examples that you are about to see…hint, hint

18. EXAMPLE : Find all vertice points, foci points, and graph the ellipse

19. a = 3 b = 4 c = Center ( h, k ) Major Axis vertices ( x, y ), ( x, y ) Minor Axis vertices ( x, y ), ( x, y ) Foci vertices ( x, y )

20. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( x, y ), ( x, y ) Minor Axis vertices ( x, y ), ( x, y ) Foci vertices ( x, y )

21. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( x, y ), ( x, y ) Minor Axis vertices ( x, y ), ( x, y ) Foci vertices ( x, y ) b > a, y axis is major

22. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( x, y ), ( x, y ) Minor Axis vertices ( x, y ), ( x, y ) Foci vertices ( x, y ) b > a, y axis is major (x) (y) The purple letters show what will be adjusted in the major and minor axis from the center

23. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( x, y ), ( x, y ) Minor Axis vertices ( x, y ), ( x, y ) Foci vertices ( x, y ) b > a, y axis is major (x) (y) The purple letters show what will be adjusted in the major and minor axis from the center Major axis – x stays the same, y is adjusted by ± b Minor axis – y stays the same, x is adjusted by ± a ±3 ±4

24. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, y ), ( 0, y ) Minor Axis vertices ( x, y ), ( x, y ) Foci vertices ( x, y ) (x) (y) Major axis – x stays the same, y is adjusted by ± b Minor axis – y stays the same, x is adjusted by ± a ±3 ±4

25. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, y ), ( 0, y ) Minor Axis vertices ( x, 0 ), ( x, 0 ) Foci vertices ( x, y ) (x) (y) Major axis – x stays the same, y is adjusted by ± b Minor axis – y stays the same, x is adjusted by ± a ±3 ±4

26. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, 4 ), ( 0, -4 ) Minor Axis vertices ( x, 0 ), ( x, 0 ) Foci vertices ( x, y ) (x) (y) Major axis – x stays the same, y is adjusted by ± b Minor axis – y stays the same, x is adjusted by ± a ±3 ±4

27. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, 4 ), ( 0, -4 ) Minor Axis vertices ( 3, 0 ), ( -3, 0 ) Foci vertices ( x, y ) (x) (y) Major axis – x stays the same, y is adjusted by ± b Minor axis – y stays the same, x is adjusted by ± a ±3 ±4

28. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, 4 ), ( 0, -4 ) Minor Axis vertices ( 3, 0 ), ( -3, 0 ) Foci vertices ( x, y ) (x) (y) - the Foci is adjusted by ± c - in this case, x stays the same, y is adjusted by ± c ( ±√7) ±3 ±4

29. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, 4 ), ( 0, -4 ) Minor Axis vertices ( 3, 0 ), ( -3, 0 ) Foci vertices ( 0, y ) (x) (y) - the Foci is adjusted by ± c - in this case, x stays the same, y is adjusted by ± c ( ±√7) ±3 ±4

30. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, 4 ), ( 0, -4 ) Minor Axis vertices ( 3, 0 ), ( -3, 0 ) Foci vertices ( 0, 0 ± √7 ) (x) (y) - the Foci is adjusted by ± c - in this case, x stays the same, y is adjusted by ± c ( ±√7) ±3 ±4

31. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, 4 ), ( 0, -4 ) Minor Axis vertices ( 3, 0 ), ( -3, 0 ) Foci vertices ( 0, 0 ± √7 ) (x) (y) ±3 ±4 To graph the Ellipse, plot your center, and your major & minor vertices, then sketch a smooth curve through your points.

32. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, 4 ), ( 0, -4 ) Minor Axis vertices ( 3, 0 ), ( -3, 0 ) Foci vertices ( 0, 0 ± √7 ) (x) (y) ±3 ±4 To graph the Ellipse, plot your center, and your major & minor vertices, then sketch a smooth curve through your points.

33. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = Center ( 0, 0 ) Major Axis vertices ( 0, 4 ), ( 0, -4 ) Minor Axis vertices ( 3, 0 ), ( -3, 0 ) Foci vertices ( 0, 0 ± √7 ) (x) (y) ±3 ±4 To graph the Ellipse, plot your center, and your major & minor vertices, then sketch a smooth curve through your points.

34. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse

35. a = 3 b = 5 c = 4 1 st find a, b, and c

36. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Next find the center… Center ( h, k ) Major Axis vertices ( x, y ), ( x, y ) Minor Axis vertices ( x, y ), ( x, y ) Foci vertices ( x, y )

37. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Next find the center… Center ( - 9, 3 ) Major Axis vertices ( x, y ), ( x, y ) Minor Axis vertices ( x, y ), ( x, y ) Foci vertices ( x, y )

38. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Next find the major / minor vertices… b 2 > a 2 so y is major, x is minor Center ( - 9, 3 ) Major Axis vertices ( x, y ), ( x, y ) Minor Axis vertices ( x, y ), ( x, y ) Foci vertices ( x, y )

39. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Major, change y by ± b Center ( - 9, 3 ) (y) Major Axis vertices ( x, y ), ( x, y ) (x) Minor Axis vertices ( x, y ), ( x, y ) (y) Foci vertices ( x, y ) Minor, change x by ± a ± 3, ±5

40. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Center ( - 9, 3 ) (y) Major Axis vertices ( - 9, 8 ), ( - 9, - 2 ) (x) Minor Axis vertices ( - 6, 3 ), ( - 12, 3 ) (y) Foci vertices ( x, y ) ± 3, ±5 Major, change y by ± b

41. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Center ( - 9, 3 ) (y) Major Axis vertices ( - 9, 8 ), ( - 9, - 2 ) (x) Minor Axis vertices ( - 6, 3 ), ( - 12, 3 ) (y) Foci vertices ( x, y ) ± 3, ±5 Minor, change x by ± a

42. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Center ( - 9, 3 ) (y) Major Axis vertices ( - 9, 8 ), ( - 9, - 2 ) (x) Minor Axis vertices ( - 6, 3 ), ( - 12, 3 ) (y) Foci vertices ( x, y ) ± 3, ±5 Foci is on major, change y by ± c

43. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Center ( - 9, 3 ) ±4 (y) Major Axis vertices ( - 9, 8 ), ( - 9, - 2 ) (x) Minor Axis vertices ( - 6, 3 ), ( - 12, 3 ) (y) Foci vertices ( - 9, 3 ± 4 ) ± 3, ±5 Foci is on major, change y by ± c

44. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Center ( - 9, 3 ) Major Axis vertices ( - 9, 8 ), ( - 9, - 2 ) Minor Axis vertices ( - 6, 3 ), ( - 12, 3 ) Foci vertices ( - 9, 3 ± 4 ) GRAPH – 1 st plot center, then plot major & minor vertices, then sketch your ellipse.

45. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Center ( - 9, 3 ) Major Axis vertices ( - 9, 8 ), ( - 9, - 2 ) Minor Axis vertices ( - 6, 3 ), ( - 12, 3 ) Foci vertices ( - 9, 3 ± 4 ) GRAPH – 1 st plot center, then plot major & minor vertices, then sketch your ellipse.

46. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 Center ( - 9, 3 ) Major Axis vertices ( - 9, 8 ), ( - 9, - 2 ) Minor Axis vertices ( - 6, 3 ), ( - 12, 3 ) Foci vertices ( - 9, 3 ± 4 ) GRAPH – 1 st plot center, then plot major & minor vertices, then sketch your ellipse.