(a) Eliminate the parameter to find a Cartesian equation of...

(a) Eliminate the parameter to find a Cartesian equation of the curve.
(b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as parameter increases.

$ x = \sqrt{t + 1} $, $ \quad y = \sqrt{t - 1} $

Key Concepts

Parametric Equations

Parametric equations describe a curve by expressing the coordinates (x and y) as functions of a parameter. They allow the representation of curves that may be difficult or impossible to express as a single Cartesian equation. This approach provides flexibility in describing curves, especially when the relation between x and y is not function-like or when handling more complex movements in the plane.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations so that the relation between x and y is expressed directly, resulting in a Cartesian equation. This process typically requires algebraic manipulation, such as solving one of the equations for the parameter and substituting into the other, thereby removing the parameter from the equations altogether.

Domain Restrictions

Domain restrictions are important when working with parametric equations because the original definitions often include inherent limitations on the values that the parameter and resulting coordinates can take. These restrictions ensure that the values under operations like square roots remain real, and they play a crucial role in accurately describing the portion of the curve represented by the parametric equations.

Sketching the Curve

Sketching the curve involves plotting the Cartesian equation or points obtained from the parametric equations on a coordinate plane. An essential aspect of sketching is indicating the direction in which the curve is traced as the parameter increases, which can be done by adding an arrow on the curve. This directional information provides insight into the dynamics of the curve, showing how the point moves as the parameter varies.

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