Exercises 1-18 give parametric equations and parameter...

Exercises $1-18$ give parametric equations and parameter intervals for the motion of a particle in the $x y$ -plane. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. (The graphs will vary with the equation used.) Indicate the portion of the graph traced by the particle and the direction of motion. $$ x=\frac{t}{t-1}, \quad y=\frac{t-2}{t+1}, \quad-1 < t < 1 $$

Exercises $1-18$ give parametric equations and parameter intervals for
the motion of a particle in the $x y$ -plane. Identify the particle's path by
finding a Cartesian equation for it. Graph the Cartesian equation. (The
graphs will vary with the equation used.) Indicate the portion of the
graph traced by the particle and the direction of motion.
$$
x=\frac{t}{t-1}, \quad y=\frac{t-2}{t+1}, \quad-1 < t < 1
$$

Key Concepts

Cartesian Equation

A Cartesian equation is an equation in x and y that represents the set of points on a curve without explicit reference to the parameter. It is useful for graphing and analyzing the geometric properties of the curve in the coordinate plane.

Domain Restrictions

Domain restrictions refer to the specified interval for the parameter, which translates into limitations on the portions of the curve that are traced. These restrictions are crucial as they define which parts of the Cartesian equation are relevant and indicate boundaries or endpoints of the path.

Parametric Equations

Parametric equations represent the coordinates of points on a curve or motion path as functions of one or more parameters. They allow the description of complex curves that may not be easily expressed as a single function y = f(x) and are particularly useful in representing dynamic motion, where the parameter often corresponds to time.

Elimination of the Parameter

Eliminating the parameter involves algebraically manipulating the parametric equations to remove the parameter, resulting in a direct relationship between x and y. This process helps convert the parametric representation into a Cartesian equation that describes the same geometric curve.

Direction of Motion

The direction of motion indicates the sequence in which the points on the curve are traced as the parameter increases. It provides insight into the dynamics of the particle’s movement along the path, such as the starting point, endpoint, and the orientation of traversal on the graph.

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