Answer
Please see the figure attached.
Work Step by Step
We have ${\bf{r}}\left( t \right) = \left( {\left| t \right| + t,\left| t \right| - t} \right)$.
So, $x = \left| t \right| + t$ and $y = \left| t \right| - t$.
We evaluate several points for the interval $ - 5 \le t \le 5$ and list them in the following table:
$\begin{array}{*{20}{c}}
t&{\left( {x,y} \right)}\\
{ - 5}&{\left( {0,10} \right)}\\
{ - 4}&{\left( {0,8} \right)}\\
{ - 3}&{\left( {0,6} \right)}\\
{ - 2}&{\left( {0,4} \right)}\\
{ - 1}&{\left( {0,2} \right)}\\
0&{\left( {0,0} \right)}\\
1&{\left( {2,0} \right)}\\
2&{\left( {4,0} \right)}\\
3&{\left( {6,0} \right)}\\
4&{\left( {8,0} \right)}\\
5&{\left( {10,0} \right)}
\end{array}$
Then, we plot the points in rectangular coordinates and join them to get the curve.
