Sketch the curve by using the parametric equations to plot...

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

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Key Concepts

Parametric Equations

Parametric equations represent curves by defining both x and y coordinates as functions of an independent parameter. This method is especially useful for describing curves that might not be functions in the traditional y = f(x) sense, allowing for more flexible and general representations of geometric shapes.

Evaluating and Plotting Points

Plotting points from parametric equations involves substituting various values of the parameter into the x and y functions to generate coordinate pairs. These points, when connected, form the curve, and selecting strategic parameter values helps capture the overall shape and key features of the graph.

Parameter Domain

The parameter domain specifies the range of the parameter values used in the equations. Paying attention to this domain is critical because it determines the portion of the curve that is drawn, ensuring that the representation accurately reflects the intended segment of the curve.

Direction of Tracing

The direction of tracing indicates how the curve is traversed as the parameter increases. Representing this direction on a sketch, typically with an arrow, provides insight into the orientation of the curve and is essential when the motion or progression along the curve is of interest.

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