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Conic sections are the curves obtained by intersecting a plane with a double-napped cone—the type of cone that looks like two ice cream cones stuck together at their tips. Depending on the angle of the plane's intersection with the cone, you can get different shapes: a circle, an ellipse, a parabola, or a hyperbola.

Imagine slicing through an ice cream cone at different angles. A straight horizontal cut gives you a circle, a slight tilt results in an ellipse, a steeper angle produces a parabola, and finally, a vertical cut that intersects both naps of the cone will leave you with a hyperbola. This in essence describes the fascinating world of conic sections!

In our exercise, after rearranging and completing the square, the given equation is recognized as that of a hyperbola. This is because it has the opposite signs for the squared terms, which is a unique property of hyperbolas in their standard form equation. When graphing, it's the interplay between these sections that helps us understand the curves' behaviors and their geometric properties.

Completing the square is a method used to transform quadratic equations into a form that makes it easier to analyze and solve them. In basic algebra, this technique allows us to tackle equations by identifying their vertex and finding whether they have maximum or minimum points.

For the equation at hand, the step of completing the square is crucial. By doing so, we rewrite the equation to make it look like the standard form of a hyperbola. This involves grouping the x's with x's and y's with y's, then adjusting the equation so that we have perfect square trinomials. With this, the equation becomes a visual and mathematical representation of a parabola or hyperbola. It becomes much simpler to identify the center, vertices, and asymptotes of the hyperbola, which are essential features for accurate graphing.

The standard equation of a hyperbola is a powerful tool in understanding its graph. In its most recognizable form, it's either \( (x-h)^2/a^2 - (y-k)^2/b^2 = 1 \) for a horizontally oriented hyperbola or \( (y-k)^2/a^2 - (x-h)^2/b^2 = 1 \) for a vertical one. Here, \( (h, k) \) represents the center; \( a \) is the distance from the center to the vertices along the axis of the hyperbola; and \( b \) is the distance from the center to the co-vertices, which are perpendicular to the vertices.

By identifying these components in the given equation, we can sketch the hyperbola's shape with precision. The standard equation tells us not only about the orientation but also explains the behavior of the hyperbola at infinity, guiding us through the asymptotes—the lines that the hyperbola approaches but never intersects. Building a strong foundation in working with the standard equation of a hyperbola is essential for mastering graphing and understanding the beautiful complexity of conic sections.