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

ELLIPSE – a conic section formed by the intersection of a right circular cone and a plane. ELLIPSE. Center is at ( 0 , 0 ). ELLIPSE. Center is at ( h , k ). ELLIPSE. Center is at ( 0 , 0 ). Center is at ( h , k ). Standard Form :. ELLIPSE - graphs. When a 2 > b 2. +y. -x. +x.

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

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

  5. ELLIPSE - graphs When a2 > b2 +y -x +x -y

  6. ELLIPSE - graphs When a2 > b2 +y Major axis -x +x -y

  7. ELLIPSE - graphs When a2 > b2 +y Minor axis Major axis -x +x -y

  8. ELLIPSE - graphs When a2 > b2 +y Minor axis Major axis -x +x Major axis vertices -y

  9. ELLIPSE - graphs When a2 > b2 +y Minor axis Major axis -x +x Major axis vertices Minor axis vertices -y

  10. ELLIPSE - graphs When a2 > b2 +y -x +x Foci -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

  11. ELLIPSE - graphs When a2 > b2 +y -x +x -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 b2 > a2 +y -x +x -y

  13. ELLIPSE - graphs When b2 > a2 +y Major axis -x +x -y

  14. ELLIPSE - graphs When b2 > a2 +y Major axis Minor axis -x +x -y

  15. ELLIPSE - graphs When b2 > a2 +y Major axis vertices -x +x Minor axis vertices -y

  16. ELLIPSE - graphs When b2 > a2 +y Foci -x +x -y

  17. When working with ellipses, we will always find the following : Center ( h , k ) a = √a2 Major Axis vertices ( x , y ), ( x , y ) b = √b2 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. EXAMPLE : Find all vertice points, foci points, and graph the ellipse 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 ) (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 = ±3 ±4 Center( 0 , 0 ) Major Axis vertices ( x , y ), ( x , y ) Minor Axis vertices ( x , y ) , ( x , y ) Foci vertices ( x , y ) (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

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

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

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

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

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

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

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

  31. EXAMPLE : Find all vertice points, foci points, and graph the ellipse a = 3 b = 4 c = ±3 ±4 Center( 0 , 0 ) Major Axis vertices ( 0 , 4 ), ( 0 , -4 ) Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 ) Foci vertices ( 0 , 0 ± √7 ) (y) (x) (y) 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 = ±3 ±4 Center( 0 , 0 ) Major Axis vertices ( 0 , 4 ), ( 0 , -4 ) Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 ) Foci vertices ( 0 , 0 ± √7 ) (y) (x) (y) 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 = ±3 ±4 Center( 0 , 0 ) Major Axis vertices ( 0 , 4 ), ( 0 , -4 ) Minor Axis vertices ( 3 , 0 ) , ( -3 , 0 ) Foci vertices ( 0 , 0 ± √7 ) (y) (x) (y) To graph the Ellipse, plotyour 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. EXAMPLE #2 : Find the center, all vertice points, foci points, and graph the ellipse a = 3 b = 5 c = 4 1st 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 Center ( h , k ) Major Axis vertices ( x , y ), ( x , y ) Minor Axis vertices ( x , y ) , ( x , y ) Foci vertices ( x , y ) Next find the center…

  37. 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 ( x , y ), ( x , y ) Minor Axis vertices ( x , y ) , ( x , y ) Foci vertices ( x , y ) Next find the center…

  38. 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 ( x , y ), ( x , y ) Minor Axis vertices ( x , y ) , ( x , y ) Foci vertices ( x , y ) Next find the major / minor vertices… b2 > a2 so y is major, x is minor

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

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

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