1. Mathematics 116 Bittinger Chapter 7 Conics

2. Mathematics 116 Conics A conic is the intersection of a plane an a double- napped cone.

3. Degenerate Conic Degenerate conic – plane passes through the vertex Point Line Two intersecting lines

4. Algebraic Definition of Conic

5. Definition of Conic Locus (collection) of points satisfying a certain geometric property.

6. Circle A circle is the set of all points (x,y) that are equidistant from a fixed point (h,k) The fixed point is the center. The fixed distance is the radius

7. Algebraic def of Circle Center is (h,k) Radius is r

8. Equation of circle with center at origin

9. Def: Parabola A parabola is the set of all points (x,y) that are equidistant from a fixed line, the directrix, and a fixed point, the focus, not on the line.

10. Standard Equation of Parabola Vertex at Origin Vertex at (0,0) Directrix y = -p Vertical axis of symmetry

11. Standard Equation of Parabola Opening left and right Vertex: (0,0O Directrix: x = -p Axis of symmetry is horizontal

12. Willa Cather – U.S. novelist (1873-1947) “The higher processes are all simplification.”

13. Definition: Ellipse An ellipse is the set of all points (x,y), the sum of whose distances from two distinct points (foci) is a constant.

14. Standard Equation of Ellipse Center at Origin Major or focal axis is horizontal

15. Standard Equation of Ellipse Center at Origin Focal axis is vertical

16. Ellipse: Finding a or b or c

17. Definition: hyperbola A hyperbola is the set of all points (x,y) in a plane, the difference whose distances from two distinct fixed points (foci) is a positive constant.

18. Hyperbola equation opening left and right centered at origin

19. Standard Equation of Hyperbola opening up and down centered at origin

20. Hyperbola finding a or b or c

21. Objective – Conics centered at origin Recognize, graph and write equations of Circle Parabola Ellipse Hyperbola –Find focal points

22. Rose Hoffman – elementary schoolteacher “Discipline is the keynote to learning. Discipline has been the great factor in my life.”

23. Mathematics 116 Translations Of Conics

24. Circle Center at (h,k)radius = r

25. Ellipse major axis horizontal

26. Ellipse major axis vertical

27. Hyperbola opening left and right

28. Hyperbola opening up and down

29. Parabola vertex (h,k) opening up and down

30. Parabola vertex (h,k) opening left and right

31. Objective Recognize equations of conics that have been shifted vertically and/or horizontally in the plane.

32. Objective Find the standard form of a conic – circle, parabola, ellipse, or hyperbola given general algebraic equation.

33. Example Determine standard form – sketch Find domain, range, focal points

34. Example - problem Determine standard form – sketch Find domain, range, focal points

36. Winston Churchill “It’s not enough that we do our best; sometimes we have to do what’s required.”