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\centerline{\textbf{\large Exam \#2 $\cdot$ Tuesday Oct. 11, 2005}}
\vspace*{.2in}
MATH 110 $\cdot$ Section 10 $\cdot$ Fall 2005  \hspace*{0.5in} Name \hrulefill
\vspace*{0.2in}
\begin{enumerate}

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\item[1.] (\textbf{5 points})  The graph of $y=f(x)$ is shown below, followed
by a transformation of $y=f(x)$.

\vspace*{2.5in}

Graph of $y=f(x)$.
\qquad
\qquad
\qquad
\qquad
\qquad
Graph of a transformation of $y=f(x)$.

\vspace*{0.1in}

Which of the following is the formula for the transformation of $y=f(x)$
shown above?

\vspace*{0.1in}

\begin{tabular}{lll}
(A) $y=f(-x) - 4 \qquad$&
(B) $y=-f(x) - 4 \qquad$&
(C) $y=-f(x - 4) \qquad$\\
(D) $y=-f(x) + 4 \qquad$&
(E) None of these
\end{tabular}

\vspace*{1.5in}

%% ----------------------------------------------------------------
\item[2.] (\textbf{5 points})  You can get the graph of $y=f(-2x)$ by
transforming the graph of $y=f(x)$ in the following way:

\vspace*{0.1in}

\begin{tabular}{lll}
(A) Shrink horizontally and reflect across the $x$-axis \\
(B) Shrink horizontally and reflect across the $y$-axis \\
(C) Stretch vertically and reflect across the $x$-axis \\
(D) Stretch vertically and reflect across the $y$-axis \\
(E) None of these
\end{tabular}

%% ----------------------------------------------------------------
\newpage
\item[3.] (\textbf{5 points})  The relation that vertically shrinks the
graph of $y=|x|$ and shifts the graph down ten units is:

\vspace*{0.1in}

\begin{tabular}{lll}
(A) $y = \frac{5}{4}|x| - 10 \qquad$ &
(B) $y = |\frac{7}{3} x - 10| \qquad$ &
(C) $y = \frac{3}{8} |x| - 10  \qquad$ \\
(D) $y = 7 |x - 10|$ &
(E) None of these
\end{tabular}

\vspace*{1.5in}

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\item[4.]  Let $r(t) = |t-1|$ and $s(t) = \sqrt{t-5}$.

\vspace*{0.1in}

(a) (\textbf{5 points})Find the composition $r(s(t))$.

\vspace*{1.5in}

(b) (\textbf{5 points})Find the domain of the composition $r(s(t))$.

\vspace*{1.5in}

%% ----------------------------------------------------------------
\item[5.] (\textbf{5 points})  Given the function $h(x) = \sqrt{x^2+1}$, find
two functions $f(x)$ and $g(x)$ such that $h(x) = f(g(x))$.  (Do not use
$f(x)=x$ or $g(x)=x$.)

\vspace*{1.5in}

%% ----------------------------------------------------------------
\newpage
\item[6.] (\textbf{10 points})  The following table lists $(x,y)$ coordinates
of four points, all of which lie on the same line.  Fill in the blanks.

\vspace*{0.2in}

\begin{tabular}{|r|r|}
\hline $x$ & $y$ \\
\hline
\hline 2 &  5.0 \\
\hline 3 &      \\
\hline   &  9.5 \\
\hline 8 & 14.0 \\
\hline
\end{tabular}

\vspace*{1.5in}

%% ----------------------------------------------------------------
\item[7.] (\textbf{5 points})  Given the following graph of a line, write an
equation for the line in standard form (i.e. $Ax + By + C$ for some $A$, $B$,
and $C$ which you are to determine).

\vspace*{1.5in}

\vspace*{1.5in}

%% ----------------------------------------------------------------
\item[8.] (\textbf{5 points})  A \emph{clich\'e} is a worn-out saying.  (For
example, math problems of the form ``One train travels south from Chicago at 50
miles per hour \ldots.'' are clich\'es.) During an all-day algebra review
session, instructor Kerl utters 5 mathematical clich\'es per hour, starting at
8 a.m.  In another room, instructor Gorlina utters 7 mathematical clich\'es per
hour, starting at 10 a.m.  At what time of day will instructor Kerl's students
have heard the same number of clich\'es as instructor Gorlina's students?

\vspace*{1.5in}

%% ----------------------------------------------------------------
\newpage
\item[9.] Let $f(x) = \frac{x^2}{8} + x - 1$.

\vspace*{0.1in}

(a) (\textbf{6 points}) Use algebraic methods to find the $x$-intercept(s),
$y$-intercept(s), and vertex.  (If there are no $x$-intercept(s), please say so,
along with a reason why.  Likewise for $y$-intercept(s) and vertex.)

\vspace*{2.25in}

(b) (\textbf{4 points}) Using the space provided, sketch the parabola defined
by $y = f(x)$.

\vspace*{2.25in}

%% ----------------------------------------------------------------
\item[10.] (\textbf{5 points})  Given the following graph of a parabola, find
an equation for the graph in general form (i.e. $y=ax^2 + bx + c$ for some
$a$, $b$, and $c$ which you are to determine).

%xxx figure here:  graph with vertex and another point labeled.

\vspace*{1.5in}

%\vspace*{1.5in}

%% ----------------------------------------------------------------
\newpage
\item[11.]  Your granddaughter picks up a rock from the surface of the moon
and throws it upward from an initial height of 1 meter with an initial velocity
of 7.2 meters per second.  The height of the rock above the lunar surface from
that time until the rock lands is described by the equation $h(t) = -0.8 t^2 +
7.2t + 1.0$.  The rock does not bounce.

\vspace*{0.1in}

(a) (\textbf{5 points})After how many seconds does the rock reach its maximum height?

\vspace*{1.5in}

(b) (\textbf{5 points})What is the maximum height the rock reaches?

\vspace*{1.5in}

(c) (\textbf{5 points})Using everything you know:  What is the height of the
rock after 12 seconds?

\vspace*{1.5in}

%% ----------------------------------------------------------------
\item[12.] (\textbf{5 points})  Which of the following can be complete graphs
of a polynomial?

\vspace*{1.5in}

\begin{tabular}{lll}
(A) 1 and 2 only \qquad\qquad &
(B) 1 and 3 only \qquad\qquad &
(D) 2 only \qquad\qquad \\
(C) 2 and 3 only &
(E) All of them \\
\end{tabular}

%% ----------------------------------------------------------------
\newpage
\item[13.] (\textbf{5 points})  Which of the following can be complete graphs
of a fourth-degree polynomial?

\vspace*{1.5in}

\begin{tabular}{lll}
(B) 1 only \qquad\qquad $\qquad$&
(C) 1 and 2 only $\qquad$&
(A) 1 and 3 only $\qquad$ \\
(E) All of them $\qquad$&
(E) None of them $\qquad$\\
\end{tabular}

\vspace*{1.5in}

%% ----------------------------------------------------------------
\item[14.] (\textbf{5 points})  Let $P(x) = 3x^3 - 21x^2 + 108$.  Given that
$6$ is a root of $P(x)$, find the other two roots, if they exist. If there are
no real roots, please say so, along with a reason why it is so.

\vspace*{2.0in}

%% ----------------------------------------------------------------
\item[15.] (\textbf{5 points})  Determine an equation for a polynomial
$P(x)$ with the following properties:

\begin{itemize}
\item The degree of $P(x)$ is 4.
\item $P(x)$ has a single root at $-2$, a double root at $1$, and a single root
at $4$.
\item The point $(-1, -4)$ lies on the graph of the polynomial.
\end{itemize}

\vspace*{1.5in}

\end{enumerate}
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\end{document}
