If you have a parametric equation grapher, graph the...

If you have a parametric equation grapher, graph the equations over the given intervals in Exercises $51-58 .$ Hyperbola branch $x=\sec t$ (enter as $1 / \cos (t) ), y=\tan t$ $($ enter as $\sin (t) / \cos (t)),$ over $$ \begin{array}{l}{\text { a. }-1.5 \leq t \leq 1.5} \\ {\text { b. }-0.5 \leq t \leq 0.5} \\ {\text { c. }-0.1 \leq t \leq 0.1}\end{array} $$

If you have a parametric equation grapher, graph the equations over
the given intervals in Exercises $51-58 .$
Hyperbola branch $x=\sec t$ (enter as $1 / \cos (t) ), y=\tan t$
$($ enter as $\sin (t) / \cos (t)),$ over
$$
\begin{array}{l}{\text { a. }-1.5 \leq t \leq 1.5} \\ {\text { b. }-0.5 \leq t \leq 0.5} \\ {\text { c. }-0.1 \leq t \leq 0.1}\end{array}
$$

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Key Concepts

Parametric Equations

Parametric equations describe curves by expressing the coordinates as functions of a separate parameter. This method allows the representation of curves that may not be easily expressed using a single Cartesian function, providing flexibility to depict a wide range of geometrical figures.

Trigonometric Functions in Parametric Representations

Using trigonometric functions (such as secant and tangent) to parametrize curves leverages their periodic and asymptotic properties. This approach is especially useful in highlighting the behavior of curves near certain critical points, such as where the functions have asymptotes or discontinuities.

Domain Restrictions in Parametric Graphing

Restricting the domain of the parameter is essential in plotting specific segments or behaviors of a curve. Domain limitations allow one to examine the curve's local properties, manage asymptotic behavior, and avoid regions where the functions may be undefined or explode.

Conic Sections - Hyperbola

A hyperbola is a conic section characterized by its open, two-branched structure and defined asymptotes. When a hyperbola is parametrized using trigonometric functions, the natural properties of these functions illuminate the hyperbola’s structure, emphasizing its branches and the behavior near its asymptotes.

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