Using an Equation Consider the equation 9 x^2+4 y^2-36 x-24...

Using an Equation Consider the equation $9 x^{2}+4 y^{2}-36 x-24 y-36=0$ (a) Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. (b) Change the 4$y^{2}$ -term in the equation to $-4 y^{2}$ Classify the graph of the new equation. (c) Change the $9 x^{2}-$ term in the original equation to 4$x^{2}$ . Classify the graph of the new equation. (d) Describe one way you could change the original equation so that its graph is a parabola.

Using an Equation Consider the equation $9 x^{2}+4 y^{2}-36 x-24 y-36=0$
(a) Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
(b) Change the 4$y^{2}$ -term in the equation to $-4 y^{2}$ Classify the graph of the new equation.
(c) Change the $9 x^{2}-$ term in the original equation to 4$x^{2}$ . Classify the graph of the new equation.
(d) Describe one way you could change the original equation so that its graph is a parabola.

Key Concepts

Discriminant of Conic Sections

The discriminant in the context of conic sections, derived from the coefficients of the general quadratic equation, provides a criterion to differentiate between the types of conics. The value of the discriminant indicates whether the equation represents a parabola, ellipse, or hyperbola, thus serving as an analytical tool in the classification process.

Conic Sections

Conic sections are the family of curves that result from intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. Each type has a unique form and set of characteristics, and their classification is based on the general quadratic equation in two variables and the relationships between the coefficients.

Completing the Square

Completing the square is an algebraic method used to rewrite quadratic expressions in a form that reveals important features of conic sections. This technique transforms a quadratic equation into a standard form by isolating squared terms, enabling easier identification of the conic's center, vertices, and axis of symmetry.

Transformation of Equations

Transformation of equations involves modifying coefficients or terms within a quadratic equation, which subsequently alters the properties and classification of its graph. Such changes, like altering the sign or magnitude of coefficients, can convert the graph from one type of conic section to another and are essential for understanding the sensitivity of conics to parameter changes.

Standard Forms of Conic Sections

Standard forms are the canonical representations of conic sections that directly display key features such as the center, axes, and orientation. By rewriting an equation into its standard form through processes like completing the square, one can readily classify the conic as a circle, ellipse, parabola, or hyperbola.

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