Is x^2=y^2-6 y+3 an equation for a parabola? Explain. |...

Name: Problem 133, Chapter 10: Intermediate Algebra for College Students
Uploaded: 2019-08-12T18:13:03-08:00
Duration: 35 s
Channel: Adam Dehollander
Description: Is x^2=y^2-6 y+3 an equation for a parabola? Explain.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas. The general quadratic equation in two variables represents one of these curves once it is transformed into a standard form, and identifying the type depends on the coefficients of the squared terms.

Parabolas

A parabola is a conic section that occurs when the plane is parallel to a generating line of the cone. Its equation typically contains only one squared variable, leading to a single axis of symmetry. Recognizing a parabola involves checking that only one variable is squared, which is not the case if both variables appear quadratically.

General Quadratic Equations in Two Variables

General quadratic equations in two variables can represent various conic sections depending on the coefficients of the squared terms. Analyzing the signs and presence of x² and y² allows one to determine whether the graph is a parabola, ellipse, or hyperbola. In the given equation, both x² and y² terms are present, which is inconsistent with the standard form of a parabola.

Completing the Square

Completing the square is a method used to rewrite quadratic expressions into a standard form that makes the features of the conic section more apparent. This technique reveals the center, vertex, and other characteristics by organizing the quadratic terms, and it is crucial in identifying the specific type of conic section represented by an equation.

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