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\centerline{MONT 106N -- Identifying Patterns}
\centerline{Group Discussion -- Lines}
\centerline{October 5, 2009}
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\noindent
{\it Background and Goals}
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After the Columbus Day break, we will be starting into
a new section of the Freedman, Pisani, Purves text
dealing with \emph{correlation and linear regression}, 
two important statistical techniques aimed at identifying
relationships between quantities.  To deal with both
of these ideas, we need to review a very basic topic from
high school algebra and geometry:  the various forms of 
equations of straight lines in the coordinate plane.
This should be something you know well (or \emph{knew well
at some point}!).  The purpose of today's discussion is
to ``jog the memory.''
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\noindent
{\it Discussion Questions}
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\begin{enumerate}
\item[A.]  We all know that given two different points $P = (x_1,y_1)$ and 
$Q = (x_2,y_2)$ in the plane there is one and only one straight line 
containing both $P$ and $Q$.
What are two other geometric ways to specify a straight line 
\emph{uniquely}?  (One of these might be a special case of the other!)
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\item[B.]  Your answers to part A lead to two different ways to write down
the $xy$-equation of the line.  What are they, and what are the corresponding
formulas?  What constants are involved here, and what do they mean geometrically?
  (Notes: You can assume that 
the $x$-coordinates of $P$ and $Q$ are different here:  $x_1 \ne x_2$.  How
does that make things simpler? What happens if $x_1 = x_2$?  How is that case 
different?)
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\item[C.]  Find an $xy$-equation for the line that is described in each part.
\begin{enumerate}
\item[1.]  The line passing through the points $P = (1,2)$ and $Q = (4,-1)$.
\item[2.]  The line passing through the point $P = (5,3)$ and \emph{parallel} to the 
line with equation $y = 2x + 4$.  
\item[3.]  The line passing through the point $P = (5,3)$ and \emph{perpendicular}
to the line from part 2.
\item[4.]  The line passing through the $y$-axis at $(0,2)$ and crossing the 
$x$ axis at an angle of $45^\circ$ (measured from the $x$-axis to your line).
\end{enumerate}
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\item[D.]  Suppose $x$ and $y$ are two quantities we can measure that are 
related in some way -- changing the value of $x$ changes the value of $y$.  Think
$x = $ temperature measured in degrees Celsius and $y = $ temperature
measured in degrees Fahrenheit, for instance.
We say that the two quantities $x,y$ are \emph{(exactly) linearly related} if
the scatter plot of all measurements of $x$ and corresponding measurements of 
$y$ would show points along
a single straight line.  
\begin{enumerate}
\item[1.]  Are $F = $ temperature measured in degrees Celsius and $C = $ temperature
measured in degrees Fahrenheit linearly related?  If so, what is the 
equation of the line that the $(F,C)$ points lie along.  (Recall that 
$32^\circ$F is the same as $0^\circ$C and $212^\circ$F is the same as $100^\circ$C.)
\item[2.]  Today's forecast high temperature in Worcester is $61^\circ$F.  What is 
the equivalent temperature in degrees Celsius?
\item[3.]  If you write your line from part 1 in the form $C = mF + b$, what do 
the constants $m,b$ represent in ``temperature terms''  (what is their
physical significance)?
\item[4.]  Is there a value of $F$ where $C = F$?  If so, what is it?
\end{enumerate}
\end{enumerate}
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\noindent
{\it Assignment}
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Group writeups due at the end of class.
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