Answer
The arrows indicate the direction of motion.
Work Step by Step
We have $x\left( t \right) = 1 + \cos t$ and $y\left( t \right) = \sin 2t$.
For the interval $0 \le \theta \le 2\pi $, we evaluate several points in rectangular coordinates corresponding to $\theta = 0,\frac{\pi }{4},\frac{\pi }{2},...,2\pi $ and list them on the following table:
$\begin{array}{*{20}{c}}
t&{\left( {x,y} \right)}\\
0&{\left( {2,0} \right)}\\
{\frac{\pi }{4}}&{\left( {1.71,1} \right)}\\
{\frac{\pi }{2}}&{\left( {1,0} \right)}\\
{\frac{{3\pi }}{4}}&{\left( {0.293, - 1} \right)}\\
\pi &{\left( {0,0} \right)}\\
{\frac{{5\pi }}{4}}&{\left( {0.293,1} \right)}\\
{\frac{{3\pi }}{2}}&{\left( {1,0} \right)}\\
{\frac{{7\pi }}{4}}&{\left( {1.71, - 1} \right)}\\
{2\pi }&{\left( {2,0} \right)}
\end{array}$
Then we plot the points in rectangular coordinates and sketch the curve by connecting these points. The arrows indicate the direction of motion as are shown in the figure.
