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\centerline{MATH 135 -- Calculus 1}
\centerline{Practice on Linear and Quadratic Functions}
\centerline{September 6, 2019}
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\noindent
{\it Background}
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Every line in the plane is described by an equation of the form $Ax + By + C = 0$ for some
constants $A,B,C$.  If the line is not vertical ($B \ne 0$), then it is the graph of a function.
(Why?)
Recall that we say a function $f$ is {\it linear} if $f(x) = mx + b$ for some constants $m, b$.
The number $m$ is called the {\it slope of the line} and the constant $b$ is called the 
{$y$-intercept} of the line.  {\it Quadratic functions} have the form $f(x) = A x^2 + B x + C$
where $A \ne 0$.
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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)] 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 $(x,y) = (1,2)$?
\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[2)]  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 a equation of a different form for $V$ as a function of $p$?
\item[(3)] By the algebraic technique of \emph{completing the square}, every quadratic function $f(x) = x^2 + Bx + C$ (note the coefficient $A = 1$) can be rewritten in the form
$$f(x) = \left(x + \frac{B}{2}\right)^2 + \frac{4C - B^2}{4}$$
\begin{enumerate}
\item What does this tell us about the relation between the graphs $y = f(x)$ and $y = x^2$?
Describe in words.
\item Now suppose $A \ne 1$.  What is true about the graph $y = A x^2 + Bx + C$ when
$A > 0$?  What about when $A < 0$?
\end{enumerate}
\end{enumerate}
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