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. $4 x^2-9 y^2-16 x+18 y-7=0$

Key Concepts

Conic Sections

Conic sections are curves that result from the intersection of a plane with a double-napped cone. The general second?degree equation in two variables represents various types of conics (circles, ellipses, parabolas, and hyperbolas) depending on the coefficients of the quadratic terms. Understanding the conic section involved helps in identifying the process to rewrite the equation in a standard form.

Completing the Square

Completing the square is an algebraic process used to convert a quadratic expression into a perfect square trinomial. This technique is essential for rewriting a conic section’s equation into its standard form, which reveals key features such as the center (or vertex) of the curve, as well as its orientation and dimensions.

Translation of Coordinate Axes

Translation of coordinate axes involves shifting the origin of the coordinate system to a new point in the plane. In the context of conic sections, this shift is often done by moving the origin to the center (or vertex) of the conic. This results in a simpler equation in the new coordinates, which makes the analysis and graphing of the conic easier.

Standard Form of Hyperbola

The standard form of a hyperbola presents the equation in a way that clearly indicates its center, transverse and conjugate axes, and asymptotes. For hyperbolas, the equation typically takes the form (x-h)²/a² - (y-k)²/b² = 1 or its variation. This form is useful for understanding the geometry of the hyperbola and for sketching its graph accurately.

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