(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.
$x=3-4 t, \quad y=2-3 t$

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Key Concepts

Parametric Equations

Parametric equations are a way to define a relationship between variables by expressing them as functions of an independent parameter. They allow for a dynamic description of curves, making it easier to capture complex motions or shapes that might be difficult to represent using a single Cartesian equation.

Elimination of the Parameter

Eliminating the parameter involves manipulating the parametric equations to remove the independent variable, resulting in a direct relationship between the dependent variables. This process converts the parametric form into a Cartesian equation, which often simplifies analysis and graphing of the curve.

Graphing Parametric Curves

Graphing a parametric curve requires computing coordinate points by substituting specific parameter values into the parametric equations and then plotting these points on the Cartesian plane. This method helps in visualizing the shape and structure of the curve, providing insight into its behavior.

Direction of Parameter Increase

The direction in which a parametric curve is traced is determined by the progression of the independent parameter. Indicating this direction is important for understanding how the curve is generated over time or as the parameter increases, clarifying the orientation and traversal of the curve.

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