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A hyperbola is a type of conic section that can be defined by its equation. The standard form of a hyperbola's equation is: either \[\begin{equation}\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \textrm{(horizontal hyperbola)}\footnote: A horizontal hyperbola opens to the left and right}\end{equation}\]or\[\begin{equation}\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\textrm{(vertical hyperbola)}\footnote: A vertical hyperbola opens upward and downward}\end{equation}\]In this exercise, the given equation is \(x^{2}-y^{2}=1\). This is a horizontal hyperbola because the \(x^2\) term is positive, and it matches the standard form:\[\begin{equation}\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\end{equation}\]Here, both \(a^2\) and \(b^2\) are equal to 1, meaning \(a = 1\) and \(b = 1\).

Asymptotes are lines that a graph approaches but never actually touches or crosses. For hyperbolas, asymptotes give us a way to understand the shape and orientation of the curve. To find the asymptotes of a hyperbola in the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), we use the equations:\[\begin{equation} y = \pm \frac{b}{a}x\end{equation}\]In the given exercise, since both \(a = 1\) and \(b = 1\), the asymptotes are:\[\begin{equation}y = \pm x\end{equation}\]These lines go through the origin and have slopes of 1 and -1, creating an 'X' shape on the graph.

Graphing a hyperbola involves several steps to ensure accuracy. Here we use the hyperbola given by the equation \( x^{2}-y^{2}=1 \).1. **Identify the center**: Since the equation does not shift horizontally or vertically, the center is at the origin, (0,0).
2. **Draw the asymptotes**: For our equation, these are \( y = x \) and \( y = -x \). Draw these asymptotes as dashed lines.
3. **Plot the vertices**: The vertices are located at (\(\pm a, 0\)), which in this case are (\( \pm 1, 0 \)) because \( a = 1 \).
4. **Sketch the hyperbola**: Use the vertices and asymptotes as guides. The hyperbola will curve around the vertices and approach the asymptotes but will never touch them.Following these steps will help you create an accurate graph of the hyperbola and understand its structure better.