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\problemgrouptitle{Parametric Equations}

\begin{problemblock}{In each of the following,
    \begin{enumerate}
    \item Sketch the graph of the curve having the indicated parametric equations.
    \item Eliminate the parameter to find a Cartesian equation of the curve.
    \end{enumerate}
  }
  
\item $x=t^3+10,\ \ y=t^3+7;\ \ \ -1\leq t\leq 2$
\answer{
  \begin{enumerate}\setcounter{enumii}{1}
  \item line segment: $y=x-3,\ \ 9\leq x\leq 18$
  \end{enumerate}
}

\item $x=t+2,\ \ y=t^{2}+4;\ \ \  t\in \mathset{R}$
\answer{
  \begin{enumerate}\setcounter{enumii}{1}
  \item parabola: $\displaystyle y=\left(x-2\right)^2+4$
  \end{enumerate}
}

\item $x=3\cos(2t),\ \ \ y=3\sin(2t);\ \ \ \ 0\leq t\leq 2\pi$
\answer{
  \begin{enumerate}\setcounter{enumii}{1}
  \item circle: $x^2+y^2=9$
  \end{enumerate}
}

\item $x=5\cos t,\ \ y=4\sin t;\ \ \ \ 0\leq t\leq 2\pi$
\answer{
  \begin{enumerate}\setcounter{enumii}{1}
  \item ellipse: $\displaystyle\frac{x^2}{25}+\frac{y^2}{16}=1$
  \end{enumerate}
}

\item $x=2\sec t+1,\ \ y=3\tan t+2;\ \ \ \ 0\leq t\leq 10$
\answer{
  \begin{enumerate}\setcounter{enumii}{1}
  \item hyperbola: $\displaystyle \frac{(x-1)^2}{4}+\frac{(y-2)^2}{9}=1$
  \end{enumerate}
}

\end{problemblock}

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