Question 142052
Identify vertex, focus, directrix, axis of symmetry and latus rectum from the following parabola equation:
{{{x=(1/8)(y+1)^2+3}}}

Two things you must know about parabolas, their graphs
and their equations 

1. The parabola whose equation is

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

opens upward if p is positive, and downward if p is negative.
It has:

vertex, the point (h,k),
focus, the point (h,k+p),
directrix, the horizontal line whose equation is y=k-p
length of latus rectum = 4p,
endpoints of the latus rectum, the points (h-2p,k+p),(h+2p,k+p)

2. The parabola whose equation is

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

opens to the right if p is positive, and 
to the left if p is negative.
It has:

vertex, the point (h,k),
focus, the point (h+p,k),
directrix, the vertical line whose equation is x=h-p
length of latus rectum = 4p,
endpoints of the latus rectum, the points (h+p,k-2p),(h+p,k+2p)

Your parabola is the second type:

{{{x=(1/8)(y+1)^2+3}}}

or

{{{x-3=(1/8)(y+1)^2}}}

Compare that to

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

{{{-h=-3}}} so {{{h=3}}}
{{{4p=1/8}}} so {{{p=1/32}}}
{{{-k=1}}} so {{{k=-1}}}

It opens to the right because {{{p=1/32}}}, a positive number.

It has:

vertex, the point (h,k) = ({{{3}}},{{{-1}}})
focus, the point (h+p,k) = ({{{3+1/32}}},{{{-1}}}) = ({{{97/32}}},{{{-1}}}) 
directrix, the vertical line whose equation is {{{x=h-p}}} or {{{x=3-1/32}}} or {{{x =95/32}}} 
length of latus rectum = {{{4p}}} = {{{4(1/32)}}} = {{{4/32}}} = {{{1/8}}}
endpoints of the latus rectum, the points ({{{h+p}}},{{{k-2p}}}) and ({{{h+p}}},{{{k+2p}}}), or ({{{97/32}}},{{{-17/16}}}) and {{{97/32}}},{{{-15/16}}})

The parabola looks like this.  The vertical line is the directrix.
The focus is the little dot just inside the parabola.  I won't try to
draw the latus rectum.  It is a very short line, only {{{1/8}}} of a
unit that goes across the parabola through the focus. 

{{{drawing(400,375,-3,7,-6,4, locate(97/32-.05,-.7,"."),
graph(400,375,-3,7,-6,4,sqrt(8x-24)-1),line(95/32,-10,95/32,10),
graph(400,375,-3,7,-6,4,-sqrt(8x-24)-1),line(97/32,-17/16-.05,97/32,-15/16+.05)  )}}} 

Edwin</pre>