91Ó°ÊÓ

Understanding the domain and range of a hyperbola is crucial for sketching its graph accurately. For the hyperbola represented by the equation \(x^2 - y^2 = 1\), the domain and range tell us where on the coordinate plane the hyperbola exists.

* **Domain**: Since the equation is \(x^2 - y^2 = 1\), only certain values of \(x\) will satisfy this equation. More specifically, \(x^2\) must be equal to or greater than 1 in order to not result in a negative when \(y^2\) is considered. Therefore, the values \(x\) can take on are outside the interval from -1 to 1. Thus, the domain of this hyperbola is \((-fty, -1) \cup (1, fty)\).
* **Range**: The range tells us the possible values for \(y\). For this hyperbola, \(y^2 = x^2 - 1\), which means \(y^2\) must be at least 0, leading to \(|y|\) values that cannot be limited in size. Consequently, the range becomes \((-fty, -1) \cup (1, fty)\). Both domain and range help us understand how widely the hyperbola stretches across the x and y-axes.

The standard form of a hyperbola forms the foundation of graphing and analyzing its properties. The given equation \(x^2 - y^2 = 1\) is already in a form that resembles the more generic hyperbola equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). This standard form makes it easy to identify key characteristics.

To match to the standard form, we compare coefficients with \(a^2\) and \(b^2\):

• Here, \(a^2 = 1\) and \(b^2 = 1\), which means both \(a\) and \(b\) equal 1.

The standard form reveals the orientation, center, and more, making it a valuable tool in graphing hyperbolas.

The asymptotes of a hyperbola provide boundaries that the curve approaches but never actually reaches. These lines are crucial in sketching the hyperbola since they give insight into its shape.

For a hyperbola in the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the asymptotes are determined by the equation \(y = \pm \frac{b}{a}x\). In this case, both \(a\) and \(b\) are 1, leading to slopes of \(\pm 1\). Thus, the asymptotes are the lines \(y = x\) and \(y = -x\).

These lines cross at the origin and help define where the branches of the hyperbola will be, making them essential components in accurately drawing these curves.

The vertices of a hyperbola are the points where the curve makes its closest approach to the center along the transverse axis. These are key landmarks in its geometry.

In the context of the equation \(x^2 - y^2 = 1\), the center of the hyperbola is clearly at the origin \((0, 0)\). With \(a = 1\), the vertices are located at \((a, 0)\) and \((-a, 0)\), which boils down to \((1, 0)\) and \((-1, 0)\) respectively. These points lie along the x-axis and mark the widest points on the hyperbola's transverse axis.

Identifying the vertices is not just about sketching the graph accurately—it's also about understanding the hyperbola's dimensions and orientation.