Conic Sections Translation of Axes:
Parabola:
The standard equation for translated parabolas is:
(x - h)<sup>2</sup> = 4p(y - k)<sup>2</sup> or (y - k)= 4p(x - h)vertex (h,k) (h,k)axis x = h y = kp is the focus of the parabola p > 0 opens up opens rightp < 0 opens down opens leftfocus (h, k + p) (h + p,k)directrix y = k - p x = h - plength of latus rectum 4|p| 4|p|endpoints of latus rectum (h ± 2p,k + p) (h + p, k ± 2p)
ex.
graph x<sup>2 </sup>- 6x - 8y + 49 = 0 convert to standard formx<sup>2 </sup>- 6x = 8y - 49 complete the square on the x termsx<sup>2 </sup>- 6x + 9 = 8y - 49 +9(x - 3)<sup>2</sup> = 8y - 40(x - 3)<sup>2</sup> = 8(y - 5) vertical parabola
vertex: (3,5)
axis: x = 3
4p = 8
p = 2 > 0; opens up
focus: (3,7)
directrix: y = 3
length of latus rectum: 4|2| = 8 (or ± 2p)
endpoints of latus rectum: (7,7) and (-1,7)
ex.
graph y<sup>2 </sup>+ 8y + 12x - 8 = 0 convert to standard formy<sup>2 </sup>+ 8y = -12x + 8 complete the square on the x termsy<sup>2</sup> + 8y + 16 = -12x + 8 +16(y + 4)<sup>2</sup> = -12x + 24(y + 4)<sup>2</sup> = -12(x - 2) horizontal parabola
vertex: (2,-4)
axis: y = -4
4p = -12
p = -3 < 0; opens left
focus: (-1,-4)
directrix: x = 5
length of latus rectum: 4|-3| = 12 (or ± 2p)
endpoints of latus rectum: (-1,2) and (-1,-10)
