Question 975216
<pre>

The trouble with parabolas is that different books use different notation.
They are all equivalent, but they are just different enough to cause confusion.
I will arbitrarily pick one of the notations.  If your book is different, just
tell me what your book gives for the standard equation for a parabola in the
thank-you note below and I'll change it on here to fit your notation. One
common one is:

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

where (h,k) = the vertex, p is the distance from center to focus, positive
if parabola opens right and negative if it opens left.

We plot the focus point and draw the the directrix, a vertical line
through x=-2 on the x-axis:

{{{drawing(5600/19,400,-4,10,-2,17,
graph(5600/19,400,-4,10,-2,17),

green(circle(4,7,0.15),circle(4,7,0.13),circle(4,7,0.11),circle(4,7,0.09),circle(4,7,0.07),circle(4,7,0.05),circle(4,7,0.03),circle(4,7,0.01),line(-2,-20,-2,20),locate(4,7,"(4,7)"))




 )}}}

The vertex is a point exactly half-way between the focus and the directrix
line. That is the point (h,k) = (1,7)

{{{drawing(5600/19,400,-4,10,-2,17,
graph(5600/19,400,-4,10,-2,17),locate(1,7,"(1,7)"),

green(circle(4,7,0.15),circle(4,7,0.13),circle(4,7,0.11),circle(4,7,0.09),circle(4,7,0.07),circle(4,7,0.05),circle(4,7,0.03),circle(4,7,0.01),line(-2,-20,-2,20),locate(4,7,"(4,7)")),
circle(1,7,0.15),circle(1,7,0.13),circle(1,7,0.11),circle(1,7,0.09),circle(1,7,0.07),circle(1,7,0.05),circle(1,7,0.03),circle(1,7,0.01)
 )}}}

p = the distance from the center to the focus is 3 units here.

The focal chord (sometimes called the latus rectum) is a line which is 4p
units long (in this case 4*3 or 12 units long) bisected at the focus.  So
it's 6 units up from the focus and 6 units down from the focus:

{{{drawing(5600/19,400,-4,10,-2,17,
graph(5600/19,400,-4,10,-2,17),

green(circle(4,7,0.15),circle(4,7,0.13),circle(4,7,0.11),circle(4,7,0.09),circle(4,7,0.07),circle(4,7,0.05),circle(4,7,0.03),circle(4,7,0.01),line(-2,-20,-2,20),locate(4,7,"(4,7)"),line(4,13,4,1)),
circle(1,7,0.15),circle(1,7,0.13),circle(1,7,0.11),circle(1,7,0.09),circle(1,7,0.07),circle(1,7,0.05),circle(1,7,0.03),circle(1,7,0.01),locate(1,7,"(1,7)")
 )}}}
  
And now we can sketch the parabola:

{{{drawing(5600/19,400,-4,10,-2,17,

graph(5600/19,400,-4,10,-2,17,sqrt(12(x-1))+7),
graph(5600/19,400,-4,10,-2,17,-sqrt(12(x-1))+7),
red(arc(10,7,18,-18,170,190)),

green(circle(4,7,0.15),circle(4,7,0.13),circle(4,7,0.11),circle(4,7,0.09),circle(4,7,0.07),circle(4,7,0.05),circle(4,7,0.03),circle(4,7,0.01),line(-2,-20,-2,20),locate(4,7,"(4,7)"),line(4,13,4,1)),
circle(1,7,0.15),circle(1,7,0.13),circle(1,7,0.11),circle(1,7,0.09),circle(1,7,0.07),circle(1,7,0.05),circle(1,7,0.03),circle(1,7,0.01),locate(1,7,"(1,7)")
 )}}}

h=1,k=7, 4p = 12, so the equation is

{{{(y-7)^2}}}{{{""=""}}}{{{12(x-1)}}}

Edwin</pre>