I. General Form of the 2^{nd }degree equation in 2 variables:
Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0
A. Test the discriminant B^{2}  4AC
If B^{2}  4AC < 0

Then Ellipse, Circle, Point, or Nothing

If B^{2}  4AC > 0

Then Hyperbola or Two Intersecting Lines

If B^{2}  4AC = 0

Then Parabola, Two Parallel Lines, One Line, or Nothing

B. If B = 0, the equation becomes Ax^{2}
+ Cy^{2} + Dx + Ey + F = 0
If A and C are both positive or both negative

Then Ellipse, Circle (if A = C), Point, or Nothing

If A and C are of opposite signs

Then Hyperbola or Two Intersecting Lines

If A = 0 or B = 0

Then Parabola, Two Parallel Lines, One Line, or Nothing

II. Translations from xyaxes to x'y'axes

x' = x  h y' = y  k

or

x = x' + h y = y' + k


III. Rotations from xyaxes to x'y'axes
