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Hyperbolic functions are a group of mathematical functions that are analogous to the trigonometric functions but based on hyperbolas instead of circles. There are three primary hyperbolic functions: hyperbolic sine (\( \sinh t \)), hyperbolic cosine (\( \cosh t \)), and hyperbolic tangent (\( \tanh t \)). These functions are defined as follows:

• \( \sinh t = \frac{e^t - e^{-t}}{2} \)
• \( \cosh t = \frac{e^t + e^{-t}}{2} \)
• \( \tanh t = \frac{\sinh t}{\cosh t} \)

Hyperbolic functions are used in many areas of mathematics, including calculus and complex analysis. They have properties similar to trigonometric functions. For instance, the identity \( \cosh^2 t - \sinh^2 t = 1 \) is a key relationship when working with these functions, similar to the Pythagorean identity \( \cos^2 t + \sin^2 t = 1 \). This identity also helps when converting between parametric and Cartesian equations, as we see in this exercise. Hyperbolic functions can describe the shape of hyperbolas, which is a fundamental concept in this problem.

A Cartesian equation is a standard form equation that describes a curve in terms of the x and y coordinates. When given parametric equations, like \( x = -\cosh t \) and \( y = \sinh t \), a typical task is to manipulate these into a Cartesian format by eliminating the parameter, here \( t \).

To do this, use identities and algebraic manipulation. For hyperbolic functions, the identity \( \cosh^2 t - \sinh^2 t = 1 \) is crucial. Given the parametric forms, we express \( \cosh t \) and \( \sinh t \) in terms of x and y: \( \cosh t = -x \) and \( \sinh t = y \). Substituting these into the identity, we derive the Cartesian equation:

• Substitute \( \cosh t = -x \) into \( \cosh^2 t = x^2 \)
• Substitute \( \sinh t = y \) into \( \sinh^2 t = y^2 \)
• Use the identity: \( x^2 - y^2 = 1 \)

Hence, the Cartesian equation \(x^2 - y^2 = 1\) represents a hyperbola centered at the origin.

Graphing hyperbolas involves understanding their shape and orientation in the Cartesian plane. The equation \( x^2 - y^2 = 1 \) defines a particular type of hyperbola, with certain characteristics based on its form.

This hyperbola is centered at the origin (0,0). Its axis of symmetry is along the x-axis, indicating it opens left and right. The standard structure of a hyperbola equation \( x^2/a^2 - y^2/b^2 = 1 \) shows that the hyperbola's transverse axis extends horizontally. For our equation where \( a^2 = 1 \) and \( b^2 = 1 \), the branches are parallel to the x-axis, positioned symmetrically about the y-axis.

To graph this type of hyperbola, follow these steps:

• Start by marking the center at the origin (0,0).
• Since \( a = 1 \) and \( b = 1 \), particularly focus on the points (\( \pm 1, 0 \)). These are near the vertices of the hyperbola along the x-axis.
• Sketch the two branches of the hyperbola.

The hyperbola consists of two curves that seemingly "mirror" each other across the y-axis. Each curve extends infinitely in the horizontal direction. By understanding the characteristics and the graph arrangement, you can visualize the role of hyperbolas in real-world contexts and mathematical applications.