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