1. March 19 th copyright2009merrydavidson Conic sections

2. Conic Sections A conic section is the intersection of a plane and a double cone.

3. Parabolas A Parabola is the set of all points in a plane that are equidistant between a fixed point (focus) and a line (directrix).

4. y = a(x – h) 2 + k focus directrix Axis of symmetry focal chord (h,k) When “a” is positive, it opens UP When “a” is negative, it opens DOWN

5. Where to find them… FOCUS: inside the parabola DIRECTRIX: outside of the parabola AOS: through the vertex, perpendicular to the directrix FOCAL CHORD (latus rectum): inside of the parabola

6. The distance from the vertex to the focus and the vertex to the directrix are equal. Length of the focal chord is

7. When graphing you MUST label… vertex focus directrix axis of symmetry focal chord

8. Ex 1:y = (x +1) 2 + 5 Vertex Opens AOS a = focal pt. end points of LR (-1,5) down x = -1 8 (-1,3) (3,3) & (-5,3) directrix y = 7

9. Steps to graph: 1)Plot the vertex. 2)Find/graph the AOS 2) Find “p”. 3) Count “p” units above the vertex and “p” units below the vertex for the focus and directrix. 4) Draw in and label the focus and directrix. 5) Calculate. Go HALF that many units to the right and left of the focus. These are the end points of the focal chord. Draw in the focal chord. 6) Now you know the stretch of the parabola, draw it in.

10. Ex 2: Vertex Opens AOS a =focal pt. directrix end points of LR (-7,3) up y = 2 x = -7 (-7,4) (-6,4) & (-9,4)

11. Ex 3a)Find the equation of the parabola with focus at (-3,-2) and THE DIRECTRIX at y = -6 Graph what you know. How far is it from the focus to the directrix? 4 Find the vertex and graph it. (-3, -4) Find “a”. directrix

12. Ex 4: Given the focus at (0,5) and the vertex at (0,1); write the equation.

13. Parabolas

14. x = a(y – k) 2 + h directrix focus Axis of symmetry focal chord (h,k) When “a” is positive, it opens RIGHT When “a” is negative, it opens LEFT

15. Vertex Opens AOS a =focus directrix endpoints of LR Ex 5: x = (y + 4) 2 - 1 (-1,- 4) right y = - 4 (2,- 4) x = - 4 (2,2),(2,-10)

16. Ex 6: x = - 3(y + 1) 2 - 3 Vertex Opens AOS a =focus directrix endpoints of LR (-3, -1) left y = -1

17. Ex 7a)Find the equation of the parabola with focus at (4,0) and THE DIRECTRIX at x = -2 Graph what you know. How far is it from the focus to the directrix? 6 Find the vertex and graph it. (1, 0) Find “a”. directrix

18. Ex 8: Change to vertex form:y = x 2 + 6x + 3

19. Ex 9: Given a graph, write the equation.

20. HW: WS 10-1