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\title{\vspace*{-1.5in}\textbf{Intro to Parametric Equations (see \S1.7)}\\[3pt]\normalsize With inspiration from Faan Tone Liu}
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\maketitle
%\underline{Math 1300-400: Quiz 1}
\lhead{Math 2300-013: Calculus II}
\chead{April 20, 2018}
\rhead{Parametric Eq. Intro}
\vspace{-.7in}
\noindent
\textbf{Key Points:}
\begin{itemize}
\item The coordinates $x$ and $y$ are each defined separately as functions of time $t$. Together, they trace out a curve in the plane. The two equations for $x$ and $y$ are called \textbf{parametric equations} and $t$ is called the \textbf{parameter}.

\item Think of a bug (or a snail!) crawling on the plane. $x = x(t)$ and $y = y(t)$ keep track of its $x$- and $y$-position over time.

\item Sometimes, we can eliminate the parameter to write an equation involving only $x$ and $y$.\\[-12pt]

Alert! This can help us to draw a graph, but loses all information about movement along the graph. (e.g. We can see the slime trail left by the snail, but we don't know how fast it went or in which direction.)
\end{itemize}


\noindent
\textbf{Examples:}\vskip12pt
\begin{enumerate}
\item \begin{enumerate}[(a)]
\item Sketch the curve defined by the parametric equations
\[
\begin{cases}
&x(t) = 2\cos(t)\\
&y(t) = 2\sin(t).
\end{cases}
\] 
Indicate with an arrow the direction in which the curve is traced as $t$ increases.\vfill\vfill
\item What happens to our picture if we make the restriction $-\pi \leq t \leq 0$?\vfill
\end{enumerate}
\newpage
\item 
\begin{enumerate}[(a)]
\item Sketch the curve defined by the parametric equations
\[
\begin{cases}
&x(t) = t^2 - 2t\\
&y(t) = t+1.
\end{cases}
\]\vfill

\item Eliminate the parameter $t$ to find an equation for the graph of this curve in terms of $x$ and $y$ only.\vfill
\end{enumerate}  


\item Find Parametric equations for a line segment from $(-2,8)$ to $(1,2)$.\vfill
\end{enumerate}
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