College Algebra, Chapter 4, 4.1, Section 4.1, Problem 32

A quadratic function $h(x) = 3 - 4x - 4x^2$.

a.) Find the quadratic function in standard form.


$
\begin{equation}
\begin{aligned}

h(x) =& 3 - 4x - 4x^2
&&
\\
\\
h(x) =& -4 (x^2 + x ) + 3
&& \text{Factor out -4 from the $x$-term}
\\
\\
h(x) =& -4 \left( x^2 + x + \frac{1}{4} \right) + 3 - (-4) \left( \frac{1}{4} \right)
&& \text{Complete the square: add } \frac{1}{4} \text{ inside the parentheses, subtract } (-4)\left( \frac{1}{4} \right) \text{ outside}
\\
\\
h(x) =& -4 \left( x + \frac{1}{2} \right)^2 + 4
&& \text{Factor and simplify}

\end{aligned}
\end{equation}
$


The standard form is $\displaystyle h(x) = -4 \left( x + \frac{1}{2} \right)^2 + 4$.

b.) Draw its graph.







c.) Find its maximum or minimum value.

Based from the graph, since the graph opens downward the maximum value of $f$ is $\displaystyle f \left( \frac{-1}{2}, 4 \right)$.