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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Practice on Quadratic Functions and Completing the Square}
\centerline{September 6, 2017}
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\noindent
{\it Background}
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Recall that (basic form of) the algebraic method of completing the square gives a way to 
rewrite any quadratic function $x^2 + b x + c$ as
$$x^2 + bx + c = (x + r)^2 + s$$
for some constants $r,s$.  If we have a quadratic function $a x^2 + b x + c$ with $a \ne 1$
(but $a \ne 0$ as well), then we can
factor out the $a$ and apply the basic form of the process. 
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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)]  For each of the following quadratic functions $f(x)$, complete the square, then use that form
to generate a sketch of the graph $y = f(x)$ by shifting and scaling the graph $y = x^2$.  
\begin{enumerate}
\item[(a)]  $f(x) = x^2 + 4x + 5$
\item[(b)]  $f(x) = 2x^2 - 5x + 2$
\end{enumerate}
\item[2)]  A ball thrown straight upward with initial velocity $96$ ft/sec has height
$y(t) = 96t - 16t^2$ (ft) after $t$ (sec).  What is the maximum height attained by 
the ball?  (Hint:  Complete the square in the formula for $y(t)$ and think similarly to 
what you did in problem 1.)
\end{enumerate}
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