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The center of a hyperbola is a crucial reference point used to understand its graph. It acts as the midpoint of a rectangle that bounds the hyperbola within the coordinate plane. The center of a hyperbola is found directly from its standard equation written in the form \[\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]For our given equation, \[\frac{(y+6)^2}{1/9} - \frac{(x-2)^2}{1/4} = 1\]The center (h, k) is extracted by comparing with the standard form, resulting in the center at (2, -6). This indicates that the hyperbola is centered 2 units to the right and 6 units downward from the origin in the coordinate plane. Understanding the center is key in sketching the hyperbola and displaying a symmetrical distribution around this point.

The vertices are points where the hyperbola intersects its transverse axis. For a vertical hyperbola, these lie along the y-axis, relative to its center. From the equation \[\frac{(y+6)^2}{1/9} - \frac{(x-2)^2}{1/4} = 1\]we identified the center as (2, -6) and calculated \[ a = \frac{1}{3} \] The vertices are found by adding and subtracting 'a' from the y-coordinate of the center for a vertically oriented hyperbola. Thus, the vertices' coordinates are

• (2, -6 + 1/3)
• (2, -6 - 1/3)

These points indicate where the branches of the hyperbola reach their closest distance apart. Placing these vertices correctly is crucial for accurately sketching the hyperbola.

The foci are special points that lie on the axis of the hyperbola. They are located inside each branch of the hyperbola. For our vertical hyperbola, these are positioned vertically relative to the center. The formula for finding the distance to these foci from the center is determined by \[ c^2 = a^2 + b^2 \] Once solved, \[ c = \frac{\sqrt{13}}{6} \]The foci, thus, are located at the coordinates (h, k ± c), resulting in the points:

• (2, -6 + \frac{\sqrt{13}}{6})
• (2, -6 - \frac{\sqrt{13}}{6})

These foci help define the hyperbola's shape because the difference in distances from any point on the hyperbola to the foci is constant. Understanding and calculating the foci aids significantly in graphing and analytical applications.

Asymptotes are diagonal lines that the branches of a hyperbola approach but never touch. They provide a guide for sketching its overall shape and orientation. The equations of the asymptotes for a hyperbola are derived from the formula \[ y = k \pm \frac{a}{b}(x-h) \] For our equation, \[\frac{(y+6)^2}{1/9} - \frac{(x-2)^2}{1/4} = 1\]we apply these values to find the asymptotes as

• \[ y = -6 + \frac{2}{3}(x-2) \]
• \[ y = -6 - \frac{2}{3}(x-2) \]

These lines provide a framework along which the hyperbola opens. They help in plotting the hyperbola efficiently by showing how it diverges away to infinity, never crossing these limiting lines. Understanding asymptotes enhances both theoretical insight and practical graphing capabilities on coordinate planes.