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A vertical hyperbola is a specific type of hyperbola that opens up and down. This is determined by its standard equation form which is \[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \]In this equation:

• The center of the hyperbola is \((h, k)\)
• The variable \(a\) represents the distance from the center to the vertices along the y-axis, indicating an up-down orientation.
• If the equation was reversed to \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\) it would signify a horizontal hyperbola.

To put it simply, if the \(y\) term leads the equation, it forms a vertical hyperbola. This concept is crucial for identifying the graph's orientation, ensuring the hyperbola is sketched correctly. Depending on the values of \(a\) and \(b\), the exact shape may vary, but the directional opening will always remain vertical.

Conic sections are figures created by the intersection of a plane with a cone. They encompass different shapes:

• Circle
• Ellipse
• Parabola
• Hyperbola

Each of these shapes has a distinct definition and equation. Hyperbolas, like the vertical hyperbola in the exercise, occur when the plane intersects both nappes or sides of the cone, resulting in two mirror-image arcs.
Hyperbolas have different properties, such as two foci and two directrices, compared to the single focus and directrix of a parabola. Understanding conic sections helps in grasping concepts of geometry, especially when relating equations to graph shapes.
When analyzing a hyperbola, recognizing it as a conic section provides a mathematical framework for understanding its features and transformations.

Asymptotes are lines that a curve approaches as it heads towards infinity. For hyperbolas, asymptotes are lines that help guide the shape and direction.
For a vertical hyperbola, the slopes of the asymptotes are determined by \(\pm \frac{a}{b} \) in the equation \[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \]Here,

• The slopes of the asymptotes become \(\pm\) the ratio of \(a\) to \(b\).
• In the provided problem, the calculated slopes are \(2\) and \(-2\).

Asymptotes provide a boundary for the approaching curve, but the curve will never actually intersect them, offering key insight when sketching the graph of the hyperbola. It's useful to draw the asymptotes first when graphing, as they set up a guideline for sketching the hyperbola's fins.

Vertices are pivotal points of a hyperbola that indicate where the curve turns and changes direction. For a vertical hyperbola, the vertices are determined by the value \(a\) in the standard equation:\[ \frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \]

• The vertices lie along the \(y\)-axis, precisely \(a\) units above and below the center \((h, k)\).

In our exercise:

• With the center at \((-1, 2)\) and \(a = 2\), the vertices are positioned at \((-1, 4)\) and \((-1, 0)\).

These points are critical in accurately sketching the hyperbola, as they provide a visual cue for the curve's maximum and minimum reaches along the axis. Always plot the vertices after establishing the center, giving the framework necessary to shape the hyperbola accurately around its center.