How to Graph a Parabolic Equation Without a Calculator?

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SUMMARY

The discussion focuses on the process of graphing the parabolic equation x² + 12xy + 36y² + 2x - 3y - 9 = 0 without using a graphing calculator. Participants confirm that the equation is indeed parabolic, as identified through the discriminant B² - 4AC. The conversation highlights the complexity of the equation, noting that the axis of symmetry does not align with the x or y axes, which complicates the graphing process. Understanding the structure of the equation is essential for accurate graphing.

PREREQUISITES
  • Understanding of quadratic equations and their properties
  • Familiarity with the discriminant formula B² - 4AC
  • Knowledge of graphing techniques for conic sections
  • Ability to manipulate and rearrange algebraic expressions
NEXT STEPS
  • Study the properties of conic sections, specifically parabolas
  • Learn how to derive the vertex and focus of a parabola
  • Explore methods for graphing equations in two variables
  • Investigate the use of algebraic techniques for factoring complex equations
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Students, educators, and anyone interested in mastering graphing techniques for quadratic equations and conic sections.

Taiki_Kazuma
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What's the process that needs to take place to be able to graph the following equation (not-using a graphing calculator):
x2 + 12xy + 36y2 + 2x - 3y - 9 = 0

I know to use the B2 - 4AC formula to identify the equation as being parabolic, though I don't understand the formula...

Looking at the equation at first, I thought you could factor it to resemble something like:
(x2 = y or y2 = x)

But, that's where my thinking ended (or maybe it never actually began...)


Thanks!
 
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It's parabolic, but the axis of symmetry is not parallel to either the x or y axis,
 

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