(a) Sketch the curve by using the parametric equations to...

(a) Sketch the curve by using the parametric equations to plot points. Indicate with an arrow the direction in which the curve is traced as $t$ increases.
(b) Eliminate the parameter to find a Cartesian equation of the
curve.
$$
x=2 t-1, \quad y=\frac{1}{2} t+1
$$

Key Concepts

Cartesian Equation

A Cartesian equation is a relation expressed solely in terms of x and y. When derived from parametric equations, it represents the same geometric curve without referencing an external parameter. This form is particularly useful for understanding the inherent properties such as the slope, intercepts, and overall shape of the curve.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of an independent parameter. This approach allows for a flexible description of curves, particularly when the relationship between x and y is not easily expressed in a traditional Cartesian form. It is fundamental in representing paths and motions where the independent variable often represents time or another changing quantity.

Eliminating the Parameter

Eliminating the parameter involves rewriting the parametric equations as a single Cartesian equation relating x and y directly. This is done by solving one of the equations for the parameter and substituting it into the other, thereby removing the parameter from the expression. This process simplifies the analysis of the curve by allowing the use of methods applicable to Cartesian equations.

Plotting Points and Curve Direction

Plotting points involves substituting specific values of the parameter into the parametric equations to obtain corresponding coordinates, thereby creating a set of points that lie on the curve. Indicating the direction of the curve is important to capture the orientation in which the curve is traced as the parameter increases, providing insight into the dynamic behavior of the curve, which is essential in contexts like motion and flow.

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