11-18 (a) Eliminate the parameter to find a Cartesian...

$11-18$
(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 the parameter increases.
$$ y=\sqrt{t+1}, \quad y=\sqrt{t-1} $$

Key Concepts

Domain Restrictions

Domain restrictions come into play when the functions involved (such as square roots, logarithms, etc.) require certain conditions to be valid. When eliminating the parameter, it is important to consider these restrictions to ensure that all values used in the Cartesian equation are allowed, which directly affects the correct sketching and interpretation of the curve.

Cartesian Equation of a Curve

A Cartesian equation expresses the relationship between the variables x and y without reference to an independent parameter. This traditional form is useful for studying geometric properties such as shape, symmetry, and intersections with axes, and it can often reveal the nature of the curve more directly.

Direction of Curve Trace

The direction of the curve as the parameter increases indicates the order in which points on the curve are visited. This information, often shown with arrows on a graph, is crucial for understanding the orientation and movement along the parametric curve, especially in contexts involving motion.

Parametric Equations

Parametric equations represent curves by expressing the coordinates as functions of a separate variable (the parameter). This approach provides a clear description of the position on the curve as the parameter changes and allows for the analysis of motion and direction along the curve.

Eliminating the Parameter

Eliminating the parameter involves solving one of the parametric equations for the parameter and substituting that expression into the other equation. This process removes the parameter, resulting in a Cartesian equation that directly relates x and y, which can simplify the understanding and analysis of the curve.

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