SOLUTION: please help me find the vertex, focus and LR in the equation : {{{ x^2+4y+8=4 }}}.

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Question 1119655: please help me find the vertex, focus and LR in the equation : +x%5E2%2B4y%2B8=4+.
Found 3 solutions by josgarithmetic, ikleyn, greenestamps:
Answer by josgarithmetic(39627) About Me  (Show Source):
You can put this solution on YOUR website!
4y=-x%5E2-8%2B4
4y=-x%5E2-4
4y%2B4=-x%5E2
4%28y%2B1%29=-x%5E2
highlight_green%28-4%28y%2B1%29=x%5E2%29

Vertex is 1 unit away from either focus or directrix. Vertex a maximum.

VERTEX ---
FOCUS ---
DIRECTRIX y=0
L.R. 4



----------------------------------------------
-4%28y%2B1%29=-%28x-0%29%5E2
vertex is at point(0,-1);
focus at (0,-2), one unit downward from vertex

Answer by ikleyn(52864) About Me  (Show Source):
You can put this solution on YOUR website!
.
I am here to fix the errors of the @josgarithmetic solution. 

4y=-x%5E2-8%2B4

4y=-x%5E2-4

y+=+-%281%2F4%29x%5E2-1


The parabola is downward. 


Vertex is a maximum. It is (0,-1).


FOCUS      is  (0,-2)      one unit down from the vertex
DIRECTRIX  is  y= 0.       one unit up   from the vertex
L.R.       is  4           the wide of the parabola at the focus level


Answer by greenestamps(13206) About Me  (Show Source):
You can put this solution on YOUR website!


A bit of further explanation of where the answers come from -- in case you aren't familiar with it....

The vertex form of the equation of a parabola that opens up or down is

y-k+=+%281%2F%284p%29%29%28x-h%29%5E2 or y+=+%281%2F%284p%29%29%28x-h%29%5E2%2Bk

In this form, the vertex is (h,k), and p is the directed distance from the directrix to the vertex and from the vertex to the focus. Furthermore, |4p| is the length of the latus rectum.

In this problem, the equation in that form is

y+=+%28-1%2F4%29%28x%5E2%29-1

or

y+=+%281%2F%28-4%29%29%28x-0%29%5E2-1

This is in vertex form; (h,k) = (0,-1), and 4p = -4 so p = -1.

Then the directed distance from the directrix to the vertex is -1, which makes the directrix y=0; and the directed distance from the vertex to the focus is -1, which makes the focus (0,-2).

And finally the length of the latus rectum is |4p| = |-4| = 4.