(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.
$x=t-1, \quad y=t^3+1, \quad-2 \leqslant t \leqslant 2$

Key Concepts

Parametric Equations

Parametric equations represent curves by expressing the coordinates as functions of a separate parameter. In this approach, x and y values are computed based on varying t, which allows for a more flexible description of curves that might be difficult to capture with a single Cartesian equation.

Elimination of the Parameter

Eliminating the parameter involves solving one of the parametric equations for the parameter and substituting this expression into the other equation. This process derives a Cartesian equation that directly relates x and y, making it easier to analyze the curve’s overall structure without reference to the parameter t.

Graphical Representation

Sketching the curve from parametric equations typically involves calculating several (x, y) points by substituting different values of the parameter. Plotting these points on the coordinate plane, along with an arrow that indicates the trajectory as the parameter increases, provides insight into the shape and orientation of the curve.

Domain Considerations

The range of the parameter t, as specified by its domain, defines the portion of the curve that is relevant for the problem. This constraint is important when both sketching the curve and eliminating the parameter to ensure that the Cartesian equation accurately represents the segment of the curve corresponding to the given parameter values.

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