GraphShading.pg

$a    = random(-4, 2);
$b    = random(1,  4);
$g    = Compute("$b + sqrt(x - $a)");
$gtex = "$b + \sqrt{" . nicestring([ 1, -$a ]) . '}';
$ans  = Compute("3*$b + 14/3");

$graph = createTikZImage();
$graph->tikzLibraries('arrows.meta');
$graph->BEGIN_TIKZ
\tikzset{>={Stealth[scale=1.5]}}
\filldraw[
    draw=LightBlue,
    fill=white,
    rounded corners=10pt,
    thick,use as bounding box
] (-6,-2) rectangle (6,8);
\begin{scope}
    \clip[rounded corners=14pt] (-6,-2) rectangle (6,8);
    \draw[ultra thin] (-6,-2) grid (6,8);
\end{scope}
\draw[->,thick] (-6,0) -- (6,0) node[above left,outer sep=3pt] {\(x\)};
\foreach \x in {-5,...,-1,1,2,...,5}
    \draw(\x,5pt) -- (\x,-5pt) node[below] {\(\x\)};
\draw[->,thick] (0,-2) -- (0,8) node[below right,outer sep=3pt] {\(y\)};
\foreach \y in {-1,1,2,...,7}
    \draw (5pt,\y) -- (-5pt,\y) node[left]{\(\y\)};
\filldraw[draw=blue,fill=blue!50!white,fill opacity=0.5]
    ({$a+1},0) -- ({$a+1},{$b+1})
    -- plot[domain={$a+1}:{$a+4},samples=100] (\x,{$b+sqrt(\x-$a)})
    -- ({$a+4},0) -- cycle;
\draw[blue,ultra thick,->]
    plot[domain={$a}:6,samples=100] (\x,{$b+sqrt(\x-$a)});
END_TIKZ

Setup

We create a randomly transformed version of sqrt(x) as the function with a reasonably straightforward answer.

The nicestring function is useful for printing polynomials in a nice way.

The graph is created with PGtikz.pl, a very robust plotting package. The shading in done with the \filldraw command with the fill opacity=0.5. Also note that one of the sides of the filled area is the square root function.

Graph Shading

Complete Code

POD for Macro Files

PG problem file

Explanation