SOLUTION: Give the vertex, focus, directrix, and the distance across parabola at focus, and then graph the parabola. y² - 8x - 4y + 12 = 0

y² - 8x - 4y + 12 = 0

Get the y's on the left:

          y² - 4y = 8x - 12

Take the coefficient of y, which is -4,
Multiply it by , getting -2.
Square -2, getting +4, so add +4 to both 
sides:

      y² - 4y + 4 = 8x - 12 + 4

Factor the left side, combine like terms 
on the right side:

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

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


           (y-2)² = 8(x-1)

Compare that to the standard equation for
a parabola that opens right or left,

           (y-k)² = 4p(x-h)  
 
We have

-k=-2 or k = 2
4p = 8 or p = 2
-h=-1 or h = 1

vertex = (h,k) = (1,2)
focus = (h+p,k) = (1+2,2) = (3,2)
directrix is the line whose equation is x = h-p or x = 1-2 or x = -1
distance across parabola at focal point = 4p = 8

We plot the vertex, the focus and the directrix:

 

Now draw a line from the focus through the vertex to the directrix:



Now draw a square with that line as the bottom side:



Draw another square with that line as the top side:



Now we sketch the parabola through the upper and lower
right corners of those squares, through the vertex.



Edwin