College Algebra, Chapter 4, 4.1, Section 4.1, Problem 30

A quadratic function $g(x) = 2x^2 + 8x + 11$.

a.) Find the quadratic function in standard form.


$
\begin{equation}
\begin{aligned}

g(x) =& 2x^2 + 8x + 11
&&
\\
\\
g(x) =& 2 (x^2 + 4x) + 11
&& \text{Factor out 2 from the $x$-term}
\\
\\
g(x) =& 2 (x^2 + 4x + 4) + 11 - (2)(4)
&& \text{Complete the square: add $4$ inside the parentheses, subtract $(2)(4)$ outside}
\\
\\
g(x) =& 2 (x + 2)^2 + 3
&& \text{Factor and simplify}

\end{aligned}
\end{equation}
$


The standard form is $g(x) = 2 (x + 2)^2 + 3$.

b.) Draw its graph.







c.) Find its maximum or minimum value.

Based from the graph, since the graph opens upward the minimum value of $f$ is $f(-2) = 3$.