Chapter 10.1, Problem 23ES

In each part, match the parametric equation with one of the curves labelled (I)-(VI), and explain your reasoning.

$\begin{array}{l}\left(\text{a}\right)x=\sqrt{t},y=\sin 3t\\ \left(\text{b}\right)x=2\cos t,y=3\sin t\\ \left(\text{c}\right)x=t\cos t,y=t\sin t\\ \left(\text{d}\right)x=\frac{3t}{1+t^3},y=\frac{3t^2}{1+t^3}\\ \left(\text{e}\right)x=\frac{t^3}{1+t^2},y=\frac{2t^2}{1+t^2}\\ \left(\text{f}\right)x=\frac{1}{2}\cos t,y=\sin 2t\end{array}$