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A rectangular equation is a traditional form of expressing a relationship between two variables, typically in the form of a function. In mathematics, this involves eliminating parameters from parametric equations to form a direct relationship between the variables involved. In the given problem, the parametric equations are \(x = t\) and \(y=\frac{1}{t}\). By solving one equation for the parameter \(t\) and substituting it into the other, we arrive at the rectangular equation \(y = \frac{1}{x}\). This expresses \(y\) solely in terms of \(x\).

Rectangular equations are simpler to analyze and work with, especially when graphing, because they eliminate the need to consider it as a function of another variable. This is particularly useful when working with Cartesian coordinate systems or when transferring an abstract concept into a more tangible form.

A graphing calculator is a powerful tool used in mathematics to visualize equations and their corresponding graphs. When dealing with complex equations or abstract parametric forms, a graphing calculator can help bring clarity by providing a visual representation.

In the exercise, once the rectangular equation \(y = \frac{1}{x}\) is derived, a graphing calculator can be used to plot the curve within the specified window of \([-6, 6]\) by \([-4, 4]\). By inputting the equation into the calculator, it graphically displays the hyperbola, showing its behavior in both the first and third quadrants.

• This visualization reinforces understanding by exhibiting asymptotic behavior, where lines appear to approach a coordinate axis.
• It allows users to verify the theoretical equation and ensure its correct visualization.
• Graphing tools also facilitate interaction with the equation, providing insights that are not immediately obvious from inspection alone.

An interval in mathematics defines the range of possible values a variable can take. This is important in defining the shape and domain of a graph, especially for parametric or rectangular equations. In this exercise, \(t\) is given within the interval \((-fty, 0) \cup (0, \infty)\), which translates to \(x = t\) having the same interval.

This particular interval indicates that the variable cannot include zero, as division by zero in \(y=\frac{1}{x} \) is undefined. Consequently, the graph depicts two separate branches of a hyperbola, avoiding the y-axis, which corresponds to \(x=0\).

• Understanding intervals helps identify which parts of a graph are valid for certain sections of the rectangular equation.
• Intervals offer information about discontinuities, as seen with the undefined point at \(x = 0\).
• Correctly interpreting intervals ensures precise graphing within designated windows, such as the provided \([-6, 6]\) by \([-4, 4]\) window.

A hyperbola is a type of smooth curve lying in a plane, defined by its equation or parametric forms. The parametric and rectangular equations given yield the hyperbola \(y = \frac{1}{x}\). This hyperbola consists of two branches, each extending infinitely, reflecting the nature of its equation and interval.

Distinguishing features of a hyperbola include:

• Asymptotes: Lines that the hyperbola approaches but never touches. Here, the x- and y-axes are asymptotes.
• Branching: Two separate branches exist due to the interval \((-fty, 0) \cup (0, \infty)\), which causes the branches to reside in the first and third quadrants.
• Symmetry: The hyperbola \(y = \frac{1}{x}\) is symmetric about both the x- and y-axis. This symmetry aids in graphing and understanding the shape's nature.

Using a graphing calculator, students can explore the complete geometry and curvature of a hyperbola, enriching their comprehension beyond basic theoretical knowledge.