An easy basic start to recognizing equations of circles is that they have both an X squared and a Y squared. However, when you start comparing this equation to other types of graphs ie parabolas,hyperbolas, etc. it can get tricky so thats just a basic start!
Heres an example of how to find points to graph on a circle:
first, subtract the 31 from both sides,
then complete the squares for X and Y
add the 1 and 49 you calculated from completing the square to the -31
Then factor the squares of x and y into an equation
you can now decipher the radius and the center point of your cirle. since r^2=81 r=9, and your center is (-1,-7) just the opposite values of your X and Y equations. once you plot your center point, you move up,down, left, and right the number of spaces the radius is. so in this case you would go up, down, left, and right nine spaces.
Hope i helped! =]