Graphing the Equation xy^2 y: A Comprehensive Guide

The equation xy^2 y represents a complex curve that can be graphed through various mathematical techniques. Understanding and graphing this equation involves recognizing it as a conic section and using coordinate transformations to simplify its form.

Step-by-Step Graphing Process

Identify Key Points: When x 0, the equation reduces to y^2 y, leading to y 0 and y 1. This gives us two key points: (0,0) and (0,1). Transform the Equation: To simplify the equation, we first need to rewrite it in a standard form. The given equation can be rewritten as:

x2 ? 2xy ? y2 0 [/itex]

We recognize this as a conic section and attempt to convert it into standard form by completing the square.

Conversion to Standard Form

The discriminant of a conic section is given by B^2 - 4AC 0, indicating a parabola or its degenerate forms. Since we encounter a non-degenerate case, we proceed with the transformation and further simplification.

x 12 x′ ? y′ [/itex] and y 12 x′ y′ [/itex]

Substituting these into the original equation, we get:

( 12 x′ ? y′ ) 2 ? 2 ( 12 x′ ? y′ ) x′ y′ ? ( x′ y′ ) 2 0 [/itex]

This simplifies to a standard form of a parabola, providing more insights into its vertex and orientation.

Graphing the Parabola

The resulting parabola can be graphed using the completed form, revealing its vertex, focus, and directrix. The axis of symmetry through the vertex is identified, and specific points such as the focus and directrix can be calculated.

Vertex and Focus Calculation

By solving the simplified equation, we determine the vertex of the parabola to be at ( 316 , 116 ) [/itex]. Further calculations provide the focus and the equation of the directrix, which guides in plotting the parabola accurately.

Directrix and Focus

The focus of the parabola is calculated to be at ( 18 , 18 ) [/itex], and a point on the directrix is at ( 14 , 0 ) [/itex]. The directrix is perpendicular to the axis of symmetry, ensuring accurate plotting.

Final Plot

A detailed plot of the parabola with its vertex, focus, and directrix can be seen in the provided graph. This helps visualize the complete shape and properties of the parabola defined by the equation xy^2 y.

Through these steps, we have successfully graphed the equation xy^2 y and understood its properties as a rotated parabola.