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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Discussion Day on Lines and Linear Functions}
\centerline{September 5, 2017}
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\noindent
{\it Background}
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Recall that we say a function $f$ is {\it linear} if $f(x) = mx + b$ for some constants $m, b$.
The name comes from the fact that the graph $y = mx + b$ is a \emph{straight line} in the plane.
The number $m$ is called the {\it slope of the line} and the constant $b$ is called the 
{$y$-intercept} of the line.  
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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)] 
\begin{enumerate}
\item[(a)] What is the equation of the line passing through the points $(1,4)$ and $(2,7)$?
\item[(b)] Sketch the line in part (a).
\item[(c)] What is the equation of the line parallel to the line from part (a) passing 
through the point $(-1,4)$?  (Hint:  What is true about the slopes of parallel lines?)
\end{enumerate}
\item[2)] Consider lines with equations of the form $2x + cy - 3 = 0$.
\begin{enumerate}
\item[(a)]  For which value of $c$ does the line contain the point $(1,2)$?
\item[(b)]  For which value of $c$ does the line have slope $-5$?
\item[(c)]  Is there any value of $c$ such that the line is horizontal? Why or why not?
\item[(d)]  For which value of $c$ is the line perpendicular to the line given by 
$5x - 3y + 1 = 0$?  (Hint:  What is true about slopes of perpendicular lines?)
\end{enumerate}
\item[3)]  The volume $V$ (in liters) of sample of 3 grams of carbon dioxide at $27$ degrees Celsius was
measured as a function of the pressure $p$ (in atmospheres) with the results in the following table:
\[
\begin{array}{ccccccc}
p & \vline & 0.25 & 1.00 & 2.50 & 4.00 & 6.00 \\ \hline
V & \vline & 6.72 & 1.68 & 0.67 & 0.42 & 0.27
\end{array}
\]
Is $V$ (approximately) a linear function of $p$?  Why or why not?  If so, find an approximate formula
$V = m p + b$. If not, can you see an equation of a different form for $V$ as a function of $p$?
\end{enumerate}
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