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The discriminant, often denoted as \(B^2 - 4AC\), is a crucial tool in classifying conic sections. When analyzing a conic equation such as \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\), the discriminant helps identify whether the graph represents an ellipse, hyperbola, or parabola. If \(B^2 - 4AC > 0\), the equation is a hyperbola; if \(B^2 - 4AC = 0\), it indicates a parabola; and if \(B^2 - 4AC < 0\), the equation is that of an ellipse.
In the given problem, since the discriminant calculation yields 1, which is greater than 0, the graph is that of a hyperbola. This simple test can save steps, helping students quickly determine the nature of a conic section.

A hyperbola is one of the distinct shapes in conic sections, characterized by two open curves. It occurs when the plane cuts both double cones in such a way that it doesn't pass through the apex.

Key features of a hyperbola include:

• Two separate branches or curves.
• Asymptotes that the branches approach but never touch.
• A central point known as the center of the hyperbola.

In the context of the equation \(xy = 8\), once simplified and rotated, it results in a hyperbola where the branches open along the new orientation of axes. Understanding these characteristics helps when considering rotation or translating hyperbolas.

The rotation of axes is a technique used to simplify conic sections, especially when eliminating the \(xy\)-term, which complicates the graph. By rotating the coordinate system, the equation can be transformed into a simpler one.
For example, the formulas:\[ x = x'\cos\theta - y'\sin\theta \] \[ y = x'\sin\theta + y'\cos\theta \] are used to switch to a rotated coordinate system. The angle \(\theta\) is chosen such that \(\tan 2\theta = \frac{B}{A-C}\). Upon applying these changes, the original conic can be expressed in its simpler form, making it easier to graph.
In the given exercise, choosing \(\theta = 45^\circ\) successfully eliminates the \(xy\)-term, transforming the equation into a standard hyperbola.

Graphing conics is simplified when the equation is in a recognizable form. Conics include circles, ellipses, parabolas, and hyperbolas. After determining the conic type using the discriminant, and possibly rotating the axes, you proceed to sketch.
The graph of a hyperbola, like \(x'^2 - y'^2 = 16\), is centered at the origin in rotated axes and exhibits a familiar form with principal and transverse axes. Each conic type has unique properties that aid in accurate rendering of the graph, such as:

• Vertices and foci help position ellipses and hyperbolas.
• Directrix and axis of symmetry guide parabolas.

Mastering these concepts can enable students to visualize mathematical problems more effectively, ultimately building a better intuitive understanding of conic sections.