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A Brief Recap: How Do You Tell What Shape It Is?

Module by: Kenny M. Felder. E-mail the author

Summary: This module summarizes the different types and formulas of conic sections in Algebra.

Table 1
If it has... Then it’s a... Example Horizontal or Vertical?
No squared terms Line 2 x + 3 y = 7 2x+3y=7  
One squared term Parabola 2 x 2 - 10 x + 7 y = 9 2 x 2 -10x+7y=9 If you have an x 2 x 2 but no y 2 y 2 , you’re a horizontal parabola. If you have a y 2 y 2 but no x 2 x 2 , vertical.
Two squared terms with the same coefficient Circle 3 x 2 + 3 y 2 + 6 x + 3 y = 2 3 x 2 +3 y 2 +6x+3y=2  
Two squared terms with different coefficients but the same sign Ellipse 2 x 2 + 3 y 2 + 6 x + 6 y = 12 2 x 2 +3 y 2 +6x+6y=12 The difference between vertical ellipses and horizontal is based on which squared term has the larger coefficient.
Two squared terms with different signs Hyperbola 3 x 2 – 3 y 2 + 6 x + 3 y = 2 3 x 2 –3 y 2 +6x+3y=2 The difference between vertical hyperbolas and horizontal is based on which squared term is positive.

Note that all of this is based only on the squared terms! The other terms matter in terms of graphing, but not in terms of figuring out what shape it is.

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