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\begin{document}
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\begin{tabular*}{6in}{c@{\extracolsep{\fill}}c@{\extracolsep{\fill}}c}
  MIDTERM 1 &\huge TQS 120  &\normalsize Practice\\
\end{tabular*}
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\vskip .5 in
NAME: \normalsize This is a sample midterm to be used for practice.  This is \emph{not} a template for the midterm that will be given in class.  Many of the questions on the Midterm will look quite different than those appearing here.

\begin{enumerate}
\item $[4]$ TRUE/FALSE: Circle T in each of the following cases if the 
statement is \emph{always} true.  Otherwise, circle F. Let $f$ be a function, and $x$, $y$, and $z$
be real numbers with $z \neq 0$.

\begin{tabular}{lll}
~\\
T &F & $\frac{3x+y}{3z}=\frac{x+y}{z}$\\
~\\
T &F & $(x+y)^2=x^2+y^2$\\
~\\
T &F & $\vert x\vert= x $\\
~\\
T &F & $\displaystyle{\frac{3+5i}{1-2i}=-\frac{7}{5}+\frac{11}{5}i}$\\
\end{tabular}
\vskip .2 in
\large
Show your work for the following problems.  The correct answer with 
no supporting work will receive NO credit (this includes multiple
choice questions).

\normalsize
\item $[3]$ Given $3(7+x)^{-2}-4=2$, solve for $x$.
\vskip 1in

\newpage

\item $[4]$ Let $f$ be the function whose graph is below:

\includegraphics[height=3in]{piecewise2}	

Estimate the following \emph{if} possible:

\begin{tabular*}{4in}{l@{\extracolsep{\fill}}l}
$f(-3)$ & $\displaystyle{\frac{f(-3)-1}{f(-1)}}$\\
\end{tabular*}
\vfill

\begin{tabular*}{4in}{l@{\extracolsep{\fill}}l}
$f(1)$ & $(f\circ f)(-2)$\\
\end{tabular*}
\vfill

\begin{tabular*}{4in}{l@{\extracolsep{\fill}}l}
$f(-1)f(2)$ & $f(0+.5)$\\
\end{tabular*}
\vfill

\begin{tabular*}{5in}{l@{\extracolsep{\fill}}c}
$\displaystyle{\frac{f(0+.5)}{.5}}$ & The average rate of change of $f$\\
& from $x=0$ to $x=.5$\\
\end{tabular*}
\vfill



\newpage
\item $[4]$ Find the domain of $g$ where $g(x)=\frac{2-\sqrt{5-2x}}{x+10}$.
\vskip 2in

\item $[4]$ Consider the points $P=(3,4)$ and $Q=(-1,-2)$.  Find the equation to a line that goes through the point $(1,1)$ and has a perpendicular slope to the line connecting $P$ and $Q$.
\vskip 2in
\begin{itemize}
\item $[1]$ What is the $y$ intercept of the line you found?
\vskip 1in
\item $[1]$ Find the zeros of the line you found above.
\vskip 1in
\end{itemize}



\newpage


\item Let $h$ be the \emph{function} \\
defined by: 

$ h(x) = \begin{cases}
   3& x\leq1\\
  2(x-2)^2 & 1<x \\
\end{cases}
$


\vspace{-1in} \hspace{2in}  \includegraphics[height=2.5in]{blank_5}	


\begin{enumerate}
\item $[3]$ Graph $h$. (Explaining graph transformations is worth partial credit.)%quiz 3 #3
\vskip .5in
\item $[]$ What are the coordinates of the vertex on the piece of the graph above that is a parabola?
\vskip .5in 
\item $[2]$ Find all possible input(s) so that $h(x)=1$.
\vskip .5in
\item What is the range $h$?
\vskip .5in
\end{enumerate}


\item $[4]$ Given that $j(x)=-3x^2+6x-2$. Write $j$ in vertex (standard) form.  


 






\newpage
\item $[4]$ Simplify the following as much as possible: $$\frac{(2x^4y^{\frac{1}{6}})^3(6xy^3)^{-3}}{8^{\frac{-2}{3}}x^4y^4}$$
\vskip 2in


\item $[3]$ Find a cubic polynomial whose graph passes through the points $(-2,0)$ and $(1,0)$ and has a root at 6.  Note: there are many correct answers possible here.

\vskip 1.5in

\item  Let $m(x)=x^3+x^2-\frac{39}{4}x+9$ and $n(x)=x+4$.   Use long division to find $D(x)$ and $R(x)$ so that $\displaystyle{\frac{m(x)}{n(x)}=D(x)+\frac{R(x)}{n(x)}}$
\vfill


\newpage
\item Let $\displaystyle{p(x)=\frac{x-5}{7x+5}+3}$.
\begin{enumerate}
\item Given that $p$ is one-to-one (ie has an inverse), find $p^{-1}$.
\vskip 2in
\item Write the expression $p(a+h)$ and simplify.
\vskip 1in
\item Write the expression $\displaystyle{\frac{p(a+h)-p(a)}{h}}$ and simplify.
\vskip 2in
\end{enumerate}

\item A rectangular box with a volume of 60 ft$^3$ has a square base.  Find a function that models its surface area $S$ in terms of the length $x$ of one side of its base.


\end{enumerate}

\end{document}
