I. General Form of the 2nd degree equation in 2 variables:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
A. Test the discriminant B2 - 4AC
| If B2 - 4AC < 0 | Then Ellipse, Circle, Point, or Nothing |
| If B2 - 4AC > 0 | Then Hyperbola or Two Intersecting Lines |
| If B2 - 4AC = 0 | Then Parabola, Two Parallel Lines, One Line, or Nothing |
B. If B = 0, the equation becomes Ax2 + Cy2 + 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 xy-axes to x'y'-axes
| x' = x - h y' = y - k |
or | x = x' + h y = y' + k |
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III. Rotations from xy-axes to x'y'-axes
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