1. Conic Sections
©Mathworld

2. Circle
©National Science Foundation

3. Circle
The Standard Form of a circle with a center at (0,0) and a radius, r, is……..
                                                                     center (0,0) radius = 2
Copyright © Oswego City School District Regents Exam Prep Center

4. Circles
The Standard Form of a circle with a center at (h,k) and a radius, r, is……..
                                                                                                                                                                          center (3,3) radius = 2
Copyright © Oswego City School District Regents Exam Prep Center

5. Parabolas
© Art Mayoff
© Long Island Fountain Company

6. What’s in a Parabola
A parabola is the set of all points in a plane such that each point in the set is equidistant from a line called the directrix and a fixed point called the focus.
Copyright © , Math Academy Online™ / Platonic Realms™.

7. Why is the focus so important?
© Jill Britton, September 25, 2003

8. Parabola
The Standard Form of a Parabola that opens to the right and has a vertex at (0,0) is……
©1999 Addison Wesley Longman, Inc.

9. Parabola
The Parabola that opens to the right and has a vertex at (0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (p,0)
This makes the equation of the directrix x = -p
The makes the axis of symmetry the x-axis (y = 0)

10. Parabola
The Standard Form of a Parabola that opens to the left and has a vertex at (0,0) is……
© Shelly Walsh

11. Parabola
The Parabola that opens to the left and has a vertex at (0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus(-p,0)
This makes the equation of the directrix x = p
The makes the axis of symmetry the x-axis (y = 0)

12. Parabola
The Standard Form of a Parabola that opens up and has a vertex at (0,0) is……
© SparkNotes LLC, All Rights Reserved

13. Parabola
The Parabola that opens up and has a vertex at (0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (0,p)
This makes the equation of the directrix y = -p
This makes the axis of symmetry the y-axis (x = 0)

14. Parabola
The Standard Form of a Parabola that opens down and has a vertex at (0,0) is……
©1999 Addison Wesley Longman, Inc.

15. Parabola
The Parabola that opens down and has a vertex at (0,0) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (0,-p)
This makes the equation of the directrix y = p
This makes the axis of symmetry the y-axis (x = 0)

16. Parabola
The Standard Form of a Parabola that opens to the right and has a vertex at (h,k) is……
© Shelly Walsh

17. Parabola
The Parabola that opens to the right and has a vertex at (h,k) has the following characteristics……..
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (h+p, k)
This makes the equation of the directrix x = h – p
This makes the axis of symmetry

18. Parabola
The Standard Form of a Parabola that opens to the left and has a vertex at (h,k) is……
©June Jones, University of Georgia

19. Parabola
The Parabola that opens to the left and has a vertex at (h,k) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (h – p, k)
This makes the equation of the directrix x = h + p
The makes the axis of symmetry

20. Parabola
The Standard Form of a Parabola that opens up and has a vertex at (h,k) is……
Copyright © Oswego City School District Regents Exam Prep Center

21. Parabola
The Parabola that opens up and has a vertex at (h,k) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (h , k + p)
This makes the equation of the directrix y = k – p
The makes the axis of symmetry

22. Parabola
The Standard Form of a Parabola that opens down and has a vertex at (h,k) is……
Copyright © Oswego City School District Regents Exam Prep Center

23. Parabola
The Parabola that opens down and has a vertex at (h,k) has the following characteristics……
p is the distance from the vertex of the parabola to the focus or directrix
This makes the coordinates of the focus (h , k - p)
This makes the equation of the directrix y = k + p
This makes the axis of symmetry

24. Ellipse © Jill Britton, September 25, 2003
Statuary Hall in the U.S. Capital building is elliptic. It was in this room that John Quincy Adams, while a member of the House of Representatives, discovered this acoustical phenomenon. He situated his desk at a focal point of the elliptical ceiling, easily eavesdropping on the private conversations of other House members located near the other focal point.

25. What is in an Ellipse?
The set of all points in the plane, the sum of whose distances from two fixed points, called the foci, is a constant. (“Foci” is the plural of “focus”, and is pronounced FOH-sigh.)
Copyright © , Math Academy Online™ / Platonic Realms™.

26. Why are the foci of the ellipse important?
The ellipse has an important property that is used in the reflection of light and sound waves. Any light or signal that starts at one focus will be reflected to the other focus. This principle is used in lithotripsy, a medical procedure for treating kidney stones. The patient is placed in a elliptical tank of water, with the kidney stone at one focus. High-energy shock waves generated at the other focus are concentrated on the stone, pulverizing it.

27. Why are the foci of the ellipse important?
St. Paul's Cathedral in London. If a person whispers near one focus, he can be heard at the other focus, although he cannot be heard at many places in between.
© Kevin Matthews and Artifice, Inc. All Rights Reserved.

28. Ellipse General Rules x and y are both squared
Equation always equals(=) 1
Equation is always plus(+)
a2 is always the biggest denominator
c2 = a2 – b2
c is the distance from the center to each foci on the major axis
The center is in the middle of the 2 vertices, the 2 covertices, and the 2 foci.

29. Ellipse
General Rules
a is the distance from the center to each vertex on the major axis
b is the distance from the center to each vertex on the minor axis (co-vertices)
Major axis has a length of 2a
Minor axis has a length of 2b
Eccentricity(e): e = c/a (The closer e gets to 1, the closer it is to being circular)

30. Ellipse
The standard form of the ellipse with a center at (0,0) and a horizontal axis is……

31. Ellipse
The ellipse with a center at (0,0) and a horizontal axis has the following characteristics……
Vertices ( a,0)
Co-Vertices (0, b)
Foci ( c,0)
© Cabalbag, Porter, Chadwick, and Liefting

32. Ellipse
The standard form of the ellipse with a center at (0,0) and a vertical axis is……

33. Ellipse
The ellipse with a center at (0,0) and a vertical axis has the following characteristics……
Vertices (0, a)
Co-Vertices ( b,0)
Foci (0, c)
© Cabalbag, Porter, Chadwick, and Liefting

34. Ellipse
The standard form of the ellipse with a center at (h,k) and a horizontal axis is……

35. Ellipse
The ellipse with a center at (h,k) and a horizontal axis has the following characteristics……
Vertices (h a , k)
Co-Vertices (h, k b)
Foci (h c , k)
©Sellers, James

36. Ellipse
The standard form of the ellipse with a center at (h,k) and a vertical axis is……

37. Ellipse
The ellipse with a center at (h,k) and a vertical axis has the following characteristics……
Vertices (h, k a)
Co-Vertices (h b , k)
Foci (h, k c)
© Joan Bookbinder

38. Hyperbola © Jill Britton, September 25, 2003
The huge chimney of a nuclear power plant has the shape of a hyperboloid, as does
the architecture of the James S. McDonnell Planetarium of the St. Louis Science Center.

39. What is a Hyperbola?
The set of all points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
Copyright © , Math Academy Online™ / Platonic Realms™.

40. Where are the Hyperbolas?
A sonic boom shock wave has the shape of a cone, and it intersects the ground in part of a hyperbola. It hits every point on this curve at the same time, so that people in different places along the curve on the ground hear it at the same time. Because the airplane is moving forward, the hyperbolic curve moves forward and eventually the boom can be heard by everyone in its path.
© Jill Britton, September 25, 2003

41. Hyperbola General Rules x and y are both squared
Equation always equals(=) 1
Equation is always minus(-)
a2 is always the first denominator
c2 = a2 + b2
c is the distance from the center to each foci on the major axis
a is the distance from the center to each vertex on the major axis

42. Hyperbola General Rules
b is the distance from the center to each midpoint of the rectangle used to draw the asymptotes. This distance runs perpendicular to the distance (a).
Major axis has a length of 2a
Eccentricity(e): e = c/a (The closer e gets to 1, the closer it is to being circular
If x2 is first then the hyperbola is horizontal
If y2 is first then the hyperbola is vertical.

43. Hyperbola General Rules
The center is in the middle of the 2 vertices and the 2 foci.
The vertices and the covertices are used to draw the rectangles that form the asymptotes.
The vertices and the covertices are the midpoints of the rectangle
The covertices are not labeled on the hyperbola because they are not actually part of the graph

44. Hyperbola
The standard form of the Hyperbola with a center at (0,0) and a horizontal axis is……

45. Hyperbola
The Hyperbola with a center at (0,0) and a horizontal axis has the following characteristics……
Vertices ( a,0)
Foci ( c,0)
Asymptotes:

46. Hyperbola
The standard form of the Hyperbola with a center at (0,0) and a vertical axis is……

47. Hyperbola
The Hyperbola with a center at (0,0) and a vertical axis has the following characteristics……
Vertices (0, a)
Foci ( 0, c)
Asymptotes:

48. Hyperbola
The standard form of the Hyperbola with a center at (h,k) and a horizontal axis is……

49. Hyperbola
The Hyperbola with a center at (h,k) and a horizontal axis has the following characteristics……
Vertices (h a, k)
Foci (h c, k )
Asymptotes:

50. Hyperbola
The standard form of the Hyperbola with a center at (h,k) and a vertical axis is……

51. Hyperbola
The Hyperbola with a center at (h,k) and a vertical axis has the following characteristics……
Vertices (h, k a)
Foci (h, k c)
Asymptotes:
©Sellers, James

52. Rotating the Coordinate Axis
© James Wilson

53. Equations for Rotating the Coordinate Axes

54. Resources
Bookbinder, John. Unit 8: Conic Sections (College Algebra Online). 2000. June 3, 2004 <
Britton, Jill. Occurrence of the Conics. September 25, 2003. June 3, 2004 <
Cabalbag, Christain, and Porter, Amanda and Chadwick, Justin and Liefting. Nick. Graphing Conic Sections (Microsoft Power Point Presentation 1997).  June3, 2004 <

55. Resources
Finney, Ross, et. al. Calculus: Graphical, Numerical, Algebraic. Scott Foresman-Addison Wesley, 1999.
Jones, June. Instructional Unit on Conic Sections. University of Georgia. 
June 3,
Mathews, Kevin. Great Buildings Online. Great Buildings. une 3, 2004 <

56. Resources Mueller, William. Modeling Periodicity .
Mayoff, Art. San Francisco and the Golden Gate Bridge. 
June 3, 2004 
Mueller, William. Modeling Periodicity . 
<
PRIME Articles. Platomic Realms. 
<

57. Resources
Quadratics. Spark Notes from Barnes and Noble. 
June 3, 2004 
<
Roberts, Donna. Mathematics A . Oswego City School District Regents Exam Prep. 
June, 3, 2004 <
Seek One Web Services, Long Island Fountain Company. <
Sellers, James, Introduction to Conics, June 8, 2004.

58. Resources Walsh, Shelly. Chapter 9 (Precalculus). June 3, 2004
   
Walsh, Shelly. Chapter 9 (Precalculus). 
June 3, 2004 
Weissteing, Eric W. "Conic Section." From MathWorld--A Wolfram Web Resource.
Wilson, James W.  CURVE BUILDING. An Exploration with Algebraic Relations University of Georgia. 
June 3,