Find an x^' y^'-equation for the graph of the given x...

find an $x^{\prime} y^{\prime}$-equation for the graph of the given $x y$-equation so that the center or, in the case of a parabola, the vertex, of the graph is at the origin in the $x^{\prime} y^{\prime}$-system. Sketch the curve, showing both sets of coordinate axes, and identify the type of curve. $4 x^2+9 y^2+8 x+36 y+4=0$

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include ellipses, parabolas, and hyperbolas. Recognizing the type of conic from its equation is key, as each type has distinct properties and standard forms that simplify analysis and graphing.

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in a way that reveals the center or vertex of a conic section. This method transforms the quadratic terms into a perfect square plus a constant, facilitating the derivation of the standard form of the conic and making it easier to identify shifts in the coordinate system.

Translation of Axes

Translation of axes involves shifting the origin of a coordinate system to simplify the equation of a conic section. By translating the xy-coordinate system such that the center (or vertex) of the conic is at the origin, the equation takes on a standard form, which makes both analysis and graphing more straightforward.

Standard Form of a Conic Section

The standard form of a conic section’s equation is a rewritten version that clearly displays key features such as its center, vertex, axis of symmetry, and orientation. This form is obtained after techniques like completing the square and coordinate translation have been applied, allowing one to easily sketch the curve and identify its type.

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