Question 715834
<pre>
x² + 4x - 8y + 36 = 0

Since the x is the variable that is squared, get all terms
in x on the left and other terms on the right:

          x² + 4x = 8y - 36

Complete the square on the left 

1. Multiply the coefficient of x, which is 4, by {{{1/2}}}, getting 2
2. Square that result, getting 4
3.  Add to both sides.

      x² + 4x + 4 = 8y - 36 + 4

Factor the left side, collect like terms on the right

   (x + 2)(x + 2) = 8y - 32

Write the left side as a perfect square.  Factor 8 out on the right

         (x + 2)² = 8(y - 4)

Compare to

         (x - h)² = 4p(y - k)

with vertex (h,k) = (-2,4)  4p = 8, so p = 2

We plot the vertex

{{{drawing(400,400,-10,10,-10,10,graph(400,400,-10,10,-10,10),
circle(-2,4,.2), locate(-3.2,3.7,"(-2,4)") )}}} 

Since p is positive, the parabola opens upward. the directrix
is p=2 units below the vertex and has the equation y = k-p or

y = 4-2 or y=2

It is the green line below

{{{drawing(400,400,-10,10,-10,10,graph(400,400,-10,10,-10,10),
circle(-2,4,.2), locate(-3.2,3.7,"(-2,4)"),
green(line(-20,2,20,2)),locate(-5,2,y=2)  )}}}

The focus is p=2 units above the vertex (h,k+p) or (-2,4+2) or (2,6) 

{{{drawing(400,400,-10,10,-10,10,graph(400,400,-10,10,-10,10),
circle(-2,4,.2), circle(-2,4,.1),locate(-3.2,3.7,"(-2,4)"),
circle(-2,6,.2), circle(-2,6,.1),locate(-3.2,7.2,"(-2,6)"),
green(line(-20,2,20,2)),locate(-5,2,y=2)  )}}}

Draw a line from the focus directly to the directrix and draw 2
squares on each side of that line, like this.

{{{drawing(400,400,-10,10,-10,10,graph(400,400,-10,10,-10,10),
circle(-2,4,.2), circle(-2,4,.1),locate(-3.2,3.7,"(-2,4)"),
circle(-2,6,.2), circle(-2,6,.1), red(line(-6,2,2,2),line(2,2,2,6),
line(2,6,-6,6),line(-6,6,-6,2),line(-2,6,-2,2),line(-6,2,2,2)),
green(line(-20,2,20,2)),locate(-5,2,y=2),locate(-3.2,7.2,"(-2,6)")  )}}}

Then sketch in the parabola through the vertex and the upper outer
corners of the two squares:

{{{drawing(400,400,-10,10,-10,10,graph(400,400,-10,10,-10,10,4+(x+2)^2/8),
circle(-2,4,.2), circle(-2,4,.1),locate(-3.2,3.7,"(-2,4)"),
circle(-2,6,.2), circle(-2,6,.1), red(line(-6,2,2,2),line(2,2,2,6),
line(2,6,-6,6),line(-6,6,-6,2),line(-2,6,-2,2),line(-6,2,2,2)),
green(line(-20,2,20,2)),locate(-5,2,y=2),locate(-3.2,7.2,"(-2,6)")  )}}}
 
Edwin</pre>