\documentclass{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[baseline=4.8cm,xscale=1,yscale=1]
\draw[dotted,gray,opacity=0.5] (-8,-1) grid[step=0.2] (12,5);
\draw[gray,opacity=0.5] (-8,-1) grid (12,5);
\draw[thick,->,>=latex] (-8,0) -- (12,0);
\draw[thick,->,>=latex] (0,-1) -- (0,5);
\clip (-8,-1) rectangle (12,5);
\draw[thick,orange] plot[domain=-8:12] (\x,\x) node[below left,rotate=45.0,scale=0.7,outer xsep=3mm] {$y=x$};
\draw[thick,red] plot[domain=-6:12,samples=100] (\x,{sqrt(\x+6)});
% Graduation des axes
\draw[thick] (-8,0.1/1) -- (-8,-0.1/1) node[below,outer sep=2mm] {-8};
\draw[thick] (-7,0.1/1) -- (-7,-0.1/1) node[below,outer sep=2mm] {-7};
\draw[thick] (-6,0.1/1) -- (-6,-0.1/1) node[below,outer sep=2mm] {-6};
\draw[thick] (-5,0.1/1) -- (-5,-0.1/1) node[below,outer sep=2mm] {-5};
\draw[thick] (-4,0.1/1) -- (-4,-0.1/1) node[below,outer sep=2mm] {-4};
\draw[thick] (-3,0.1/1) -- (-3,-0.1/1) node[below,outer sep=2mm] {-3};
\draw[thick] (-2,0.1/1) -- (-2,-0.1/1) node[below,outer sep=2mm] {-2};
\draw[thick] (-1,0.1/1) -- (-1,-0.1/1) node[below,outer sep=2mm] {-1};
\draw[thick] (1,0.1/1) -- (1,-0.1/1) node[below,outer sep=2mm] {1};
\draw[thick] (2,0.1/1) -- (2,-0.1/1) node[below,outer sep=2mm] {2};
\draw[thick] (3,0.1/1) -- (3,-0.1/1) node[below,outer sep=2mm] {3};
\draw[thick] (4,0.1/1) -- (4,-0.1/1) node[below,outer sep=2mm] {4};
\draw[thick] (5,0.1/1) -- (5,-0.1/1) node[below,outer sep=2mm] {5};
\draw[thick] (6,0.1/1) -- (6,-0.1/1) node[below,outer sep=2mm] {6};
\draw[thick] (7,0.1/1) -- (7,-0.1/1) node[below,outer sep=2mm] {7};
\draw[thick] (8,0.1/1) -- (8,-0.1/1) node[below,outer sep=2mm] {8};
\draw[thick] (9,0.1/1) -- (9,-0.1/1) node[below,outer sep=2mm] {9};
\draw[thick] (10,0.1/1) -- (10,-0.1/1) node[below,outer sep=2mm] {10};
\draw[thick] (11,0.1/1) -- (11,-0.1/1) node[below,outer sep=2mm] {11};
\draw[thick] (12,0.1/1) -- (12,-0.1/1) node[below,outer sep=2mm] {12};
\draw[thick] (0.1/1,-1) -- (-0.1/1,-1) node[left] {-1};
\draw[thick] (0.1/1,1) -- (-0.1/1,1) node[left] {1};
\draw[thick] (0.1/1,2) -- (-0.1/1,2) node[left] {2};
\draw[thick] (0.1/1,3) -- (-0.1/1,3) node[left] {3};
\draw[thick] (0.1/1,4) -- (-0.1/1,4) node[left] {4};
\draw[thick] (0.1/1,5) -- (-0.1/1,5) node[left] {5};
\draw[dashed,purple] (10,0) node[below,scale=0.5,outer ysep=3mm] {$u_{0}$} -- (10,4.0);
\draw[dashed,purple] (10,4.0) -- (4.0,4.0);
\draw[dashed,purple] (4.0,4.0) -- (4.0,0);
\draw[dashed,purple] (4.0,0) node[below,scale=0.5,outer ysep=3mm] {$u_{1}$} -- (4.0,3.1622776601683795);
\draw[dashed,purple] (4.0,3.1622776601683795) -- (3.1622776601683795,3.1622776601683795);
\draw[dashed,purple] (3.1622776601683795,3.1622776601683795) -- (3.1622776601683795,0);
\draw[dashed,purple] (3.1622776601683795,0) node[below,scale=0.5,outer ysep=3mm] {$u_{2}$} -- (3.1622776601683795,3.0269254467476365);
\draw[dashed,purple] (3.1622776601683795,3.0269254467476365) -- (3.0269254467476365,3.0269254467476365);
\draw[dashed,purple] (3.0269254467476365,3.0269254467476365) -- (3.0269254467476365,0);
\draw[dashed,purple] (3.0269254467476365,0) node[below,scale=0.5,outer ysep=3mm] {$u_{3}$} -- (3.0269254467476365,3.0044842230818314);
\draw[dashed,purple] (3.0269254467476365,3.0044842230818314) -- (3.0044842230818314,3.0044842230818314);
\draw[dashed,purple] (3.0044842230818314,3.0044842230818314) -- (3.0044842230818314,0);
\node[below,scale=0.5,outer ysep=3mm,purple] at (3.0044842230818314,0) {$u_{4}$};
%
\end{tikzpicture}
\end{document}