SOLUTION: Identify the conic, write its equation in standard form, and give the vertex/vertices and focus/foci. x^2+4x-8y+36=0

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 , 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

 

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



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



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



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


 
Edwin