91Ó°ÊÓ

The standard form of a hyperbola is a mathematical expression that makes it easy to understand key features of the hyperbola, such as its orientation and dimensions. A hyperbola can open either horizontally or vertically, and its standard form reflects this orientation.

For a hyperbola that opens horizontally, the standard form is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
For those that open vertically, it switches to:\[ \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \]

In our given example, we transformed the equation \(144 x^{2} - 49 y^{2} = 7056\) into its standard form:\[ \frac{x^2}{49} - \frac{y^2}{144} = 1 \]
This tells us that the hyperbola is horizontal because the \(x^2\) term comes first.

By simplifying the coefficients, this form also allows us to identify the semi-major axis (\(a\)) and semi-minor axis (\(b\)). Here, \(a = 7\) and \(b = 12\), which are essential for graphing and understanding the hyperbola's shape and dimensions.

Asymptotes in a hyperbola are straight lines that the curve approaches but never actually intersects. They give the hyperbola its distinctive shape, guiding the curvature of the hyperbolic branches. Understanding asymptotes is crucial when sketching hyperbolas.

For a horizontal hyperbola like \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the equations for the asymptotes are:\[ y = \pm \frac{b}{a}x \]
This represents two diagonal lines crossing the center of the hyperbola.

In our case, with \(a = 7\) and \(b = 12\), these asymptotes are:\[ y = \pm \frac{12}{7}x \]
These lines will never be touched by the hyperbola's arms, yet they are key in shaping its direction and the width of each opening. Plotting these lines makes it easier to sketch the hyperbola by providing a visual guide for how it curves.

The domain and range of a hyperbola describe the allowable values of \(x\) and \(y\) that the hyperbola can take. This helps define the extent of the hyperbola on a coordinate plane.

- **Domain**: For a horizontal hyperbola, the domain is typically all real numbers except for a limited interval between the vertices. It can be expressed as: \[ x \leq -a \text{ or } x \geq a \]
In our example, because the vertices are located at \((-7, 0)\) and \((7, 0)\), the domain is: \[ x \leq -7 \text{ or } x \geq 7 \]
- **Range**: The range encompasses all real numbers since the arms of the hyperbola extend infinitely in the vertical direction. Thus, for our hyperbola: \[ y \in \mathbb{R} \]
Understanding the domain and range helps complete the mental map of the hyperbola's shape and the space it occupies on a plane.

The vertices of a hyperbola are critical points that represent where the hyperbola is closest to its center. These points lie on the main axis and help define the shape and location of the hyperbola.

For a horizontal hyperbola, the vertices are found along the x-axis, at points \( (a, 0) \) and \( (-a, 0) \). In the formula \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), \(a\) determines the distance to each vertex from the center.

In our example, with \(a = 7\), the vertices are:\[ (7, 0) \text{ and } (-7, 0) \]

These are the points where the hyperbola intersects its axis of symmetry, marking the path of its nearest approach. Knowing the vertices assists in sketching the correct shape and size of the hyperbola on a graph.