Graphing a Plane Curve

Graph the plane curve represented by the parametric equations \[ \begin{equation*} x( t) =3t^{2} \qquad y( t) =2t\;-2\leq t\leq 2 \end{equation*} \]

Indicate the orientation of the curve.

Solution Corresponding to each number \(t\), \(-2\leq t\leq 2,\) there are a number \(x\) and a number \(y\) that are the coordinates of a point \(( x,y) \) on the curve. We form a table listing various choices of the parameter \(t\) and the corresponding values for \(x\) and \(y\), as shown in Table 1.

The motion begins when \(t=-2\) at the point \(( 12,-4) \) and ends when \(t=2\) at the point \(( 12,4)\). Figure 2 illustrates the plane curve whose parametric equations are \(x( t) =3t^{2}\) and \(y( t) =2t\). The arrows indicate the orientation of the plane curve for increasing values of the parameter \(t\).