91Ó°ÊÓ

The eccentricity of a hyperbola is a measure of how 'stretched' it is. It's denoted by the symbol e and is always greater than 1. The formula for eccentricity in the context of hyperbolas is:
\[ e = \sqrt{1 + \frac{b^{2}}{a^{2}}} \]
Here, a and b are the distances corresponding to the real and imaginary axes, respectively. In the given problem, we have a = 1 and b = 1, giving us an eccentricity of:
\[ e = \sqrt{1 + \frac{1}{1}} = \sqrt{2} \]
This suggests that the hyperbola is moderately elongated. The greater the eccentricity, the more 'open' the branches of the hyperbola will be.

Completing the square is a method used to rewrite quadratic equations in a way that makes them easier to analyze, especially for conic sections. For the given equation:
\[ y^{2} - x^{2} + 2y - 2x - 1 = 0 \]
We group and rearrange terms to make it easier to complete the square:
\[ y^{2} + 2y - x^{2} - 2x = 1 \]
Now, focus on the y-terms and x-terms individually:
\[ y^{2} + 2y = (y + 1)^{2} - 1 \]
\[ x^{2} + 2x = (x + 1)^{2} - 1 \]
Substituting these back into the equation, we get:
\[ (y + 1)^{2} - 1 - (x + 1)^{2} + 1 = 1 \]
Simplifying, it becomes:
\[ (y + 1)^{2} - (x + 1)^{2} = 1 \]
Thus, we have transformed the original equation into the standard form of the hyperbola.

Asymptotes are straight lines that a hyperbola approaches but never touches as it extends to infinity. The equations of the asymptotes are very useful in graphing a hyperbola. For a hyperbola in the standard form
\[ \frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1 \]
the asymptotes are given by:
\[ y - k = \pm \frac{a}{b} (x - h) \]
In our case, the center of the hyperbola is (-1, -1), and the slopes are each ±1:
\[ y + 1 = \pm (x + 1) \]or equivalently:
\[ y = x \]
and
\[ y = -x - 2 \]
These lines give a framework for sketching the hyperbola and showing how it branches outward.

The foci (plural of focus) of a hyperbola are two fixed points located on the major axis. The distance between the foci \[2c \] is greater than the distance between the vertices. The formula for the distance from the center to each focus is:
\[ c = \sqrt{a^{2} + b^{2}} \]
For our hyperbola, with a = 1, and b = 1, we get:
\[ c = \sqrt{1 + 1} = \sqrt{2} \]
Using the center \((-1, -1)\), the coordinates of the foci are calculated as:
\[ (h, k + c) = (-1, -1 + \sqrt{2}) \]
and
\[ (h, k - c) = (-1, -1 - \sqrt{2}) \]
These points are crucial for understanding the shape and orientation of the hyperbola.

A hyperbola can be expressed in a standard form that reveals its geometric properties. The standard form of a hyperbola aligned along the axes is:
\[ \frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1 \]
The hyperbola can also be expressed with x and y switched, for a different orientation:
\[ \frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1 \]
Here, a represents the distance from the center to the vertices along the major axis, and b represents the distance along the minor axis. The centered equation helps determine essential properties like:

• Center \((h, k)\)
• Vertices
• Foci
• Asymptotes

For our problem, understanding how to rewrite the given equation in standard form allowed us to extract all these essential characteristics and effectively sketch the hyperbola.