Expert: Sherman D. - 12/2/2004

Question
1) the parabola y^2=8x is shifted down 2 units and right 1 unit to generate the parabola (y+2)^2= 8(x-1).
  a) find the new parabola's vertex, focus, and directrix.

2) this gives the equation for parabola and tells how many units up or down and to left and right the parabola is to be shifted. find the equation for the new parabola, and find the new vertex, focus, and directrix.

x^2=8y right 1, down 7

Answer
1)
a.)
(y + 2)^2 = 8(x - 1)

(y - k)^2 = 4p(x - h)

vertex = (h,k)
vertex = (1,-2)

4p = 8
p = 2

Directrix :
x = h - p
x = 1 - 2
x = -1

Focus :
((h + p),k)
h + p = 1 + 2 = 3
k = -2
(3,-2)

Info found at www.edison.edu/faculty/manacheril/mac_2312/webmac2312/Lesson%2025.htm

If you go to the site and look for Example #2, you will notice they have 4p = 8, p = 2, but if you read on, in the directrix formula x = h - p, you will notice that they have p to be -2 instead of 2.

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2.)
x^2 = 8y
(x - 1)^2 = 8(y + 7)

(x - h)^2 = 4p(y - k)

vertex = (h,k)
vertex = (1,-7)

4p = 8
p = 2

Directrix :
y = k - p
y = -7 - 2
y = -7 + (-2)
y = -9

Focus :
(h,(k + p))
k + p = -7 + 2 = -5
(1,-5)

Same site, but look for Example #1.
If you notice, on the graph, the have the x and y values backwards. For ex: the say the vertex is (1,-2), but on the graph, it has (-2,1)