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The discriminant is a key concept when it comes to determining the type of conic section represented by a quadratic equation. For conic sections described by equations in the form:\[ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \]the discriminant \( \Delta \) helps us classify the conic. It is calculated as:\[ \Delta = B^2 - 4AC \]Understanding the result:

• If \( \Delta > 0 \), the conic is a hyperbola.
• If \( \Delta = 0 \), the conic is a parabola.
• If \( \Delta < 0 \), the conic is an ellipse (which includes circles).

In our exercise, with \( A = 1 \), \( B = -2 \), and \( C = 1 \), we find that \( \Delta = 0 \). This immediately informs us that the equation describes a parabola.Remember, the discriminant provides a quick and useful "snapshot" of what the graph of the equation will look like, without having to graph it first.

A parabola is one of the simplest conic sections, characterized by its distinct U-shape (or reversed), and it can often take a sideways orientation as well. In its most basic form, a parabola is described by the equation:\[ y = ax^2 + bx + c \]However, in our rotated context, the equation derived was:\[ Y^2 = \sqrt{2}X + 2 \]This variation signals the parabola opens sideways. Here’s why parabolas are unique:- **Vertex:** The parabola's turning point. For the given equation, once in standard form \( Y^2 = \sqrt{2}X + 2 \), the vertex is at \( \left(-\frac{2}{\sqrt{2}}, 0\right) \) in the \( X, Y \) coordinate system.- **Axis of Symmetry:** For a sideways parabola, the axis is a horizontal line parallel to the \( X \) axis.- **Focus and Directrix:** These define the "U" or sideways shape of the parabola, dictating its width and orientation.Parabolas have applications ranging from physics trajectories to architectural structures, making them a significant topic in mathematics.

Rotating the axes is a crucial technique in adapting equations to more manageable forms. It allows us to eliminate the \( Bxy \) term, simplifying our conic section. This is particularly helpful in visualizing and graphing it.The transformation we used is defined by:\[ X = x \cos(\theta) + y \sin(\theta) \]\[ Y = -x \sin(\theta) + y \cos(\theta) \]With \( \theta = \frac{\pi}{4} \), note that \( \cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \). This rotation by 45 degrees angles the axes, mixing and balancing the \( X, Y \) terms in the equation.Steps involved in rotation:

• Substitute \( x \) and \( y \) in terms of \( X \) and \( Y \).
• Expand and simplify the resulting equation.
• Eliminate any \( Bxy \) term if any, thus achieving a simpler conic form.

This technique is foundational in graphing problems, offering a clearer and often more straightforward depiction of the conic’s properties.

Graphing conic sections usually involves depicting the unique curves that each conic (ellipse, parabola, hyperbola, or circle) presents. For our transformed and rotated parabola, graphing follows a particular method.The parabola we derived, \( Y^2 = \sqrt{2}X + 2 \), opens sideways. Here’s how to graph it:- **Plot the Axes:** Begin with the rotated axes, accounting for the 45-degree rotation (\( \theta = \frac{\pi}{4} \)). Visualize these axes tilted relative to the original \( x, y \) axes.- **Key Points:** Include the vertex \( \left(-\frac{2}{\sqrt{2}}, 0\right) \) as it’s the turning point of the curve. Focus on the axis of symmetry parallel to the \( X \) axis.- **Draw the Curve:** From the vertex, draw the parabolic arc extending on either side along the \( Y \) terms, mimicking the typical U or arch shape but oriented horizontally.- **Verification:** Check specific points by substituting \( X \) values into the equation to ensure the plotted points lie correctly on the curve.Graphing conics post-rotation provides a comprehensive visual understanding, illustrating the parabolas' trajectory and symmetry clearly.