(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 the parameter increases.
$$
x=\cos \theta, \quad y=\sec ^{2} \theta, \quad 0 \leqslant \theta<\pi / 2
$$

Key Concepts

Parametric Equations

Parametric equations represent a curve by expressing its coordinates as functions of a separate parameter. This method allows for representing more complex curves that may not be easily expressed using a single Cartesian equation.

Elimination of the Parameter

This concept involves removing the parameter from the given equations to obtain a direct relationship between the variables. It typically entails solving one of the equations for the parameter and substituting it into the other, allowing the curve to be described in Cartesian form.

Cartesian Equation

A Cartesian equation expresses the relationship between the x and y coordinates without reference to an external parameter. This form can simplify analysis and visualization of the curve's properties such as intercepts, asymptotes, and overall shape.

Curve Sketching and Orientation

Sketching a curve involves plotting its points as determined by the Cartesian equation and understanding its overall shape, while orientation refers to the direction in which the curve is traced as the parameter increases. This is important for understanding the dynamics of the curve's traversal and behavior over specific intervals.

Domain Restrictions

The domain restrictions imposed on the parameter affect the range of values for the x and y coordinates in the Cartesian equation. Attention to these restrictions is crucial for accurately representing the complete segment of the curve as defined by the original parametric equations.

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