College Algebra, Chapter 3, 3.5, Section 3.5, Problem 50

Below are the graphs of $f$ and $g$. Find a formula for the function $g$.







Based from the graph, the graph for the function $g$ can be obtained by shifting the graph of $f(x) = x^2$ two units to the right, then reflect about $x$-axis and then the result is shifted one unit upward. Thus,


$
\begin{equation}
\begin{aligned}

a(x) =& (x - 2)^2
&& \text{shifting two units to the right}
\\
\\
b(x) =& - (x - 2)^2
&& \text{reflecting about $x$-axis}
\\
\\
g(x) =& 1 - (x - 2)^2
&& \text{shift 1 unit upward}

\end{aligned}
\end{equation}
$