You can use decoration={brace, amplitude=5pt}
from decorations.pathreplacing
library of tikz.
\documentclass[tikz]{standalone}
\usetikzlibrary{decorations.pathreplacing}
\begin{document}
\begin{tikzpicture}
\draw[dashed, thin,gray!50] (0,0) grid (12,12);
\draw[very thick, -latex](-1,0) -- (12,0);
\draw[very thick, -latex](0,-1) -- (0,12);
\draw[thick, ](0,0) -- (11,9);
\draw[very thick, ](8.5,6.95) -- (8.5,3);
\draw[very thick,rotate=0,decorate, decoration={brace, amplitude=5pt}] (8.7,6.95)--(8.7,3); %%% <----Puts the vertical brace
\draw[very thick, ](8.5,3) -- (3.7,3);
\draw[very thick,rotate=0,decorate, decoration={brace, amplitude=5pt}] (8.5,2.8) -- (3.7,2.8); %%% Puts the horizontal brace.
\node at (9.4,5) {\Large $\Delta y$};
\node at (6.1,2.36) {\Large $\Delta x$};
\node at (3.5,3.2) {\Large A};
\node at (8.4,7.25) {\Large B};
\node at (8.7,2.7) {\Large C};
\foreach \x in {1,2,...,11} { \draw[thick] (\x,0) -- (\x,-0.2); }
\foreach \y in {1,2,...,11} { \draw[thick] (0,\y) -- (-0.2,\y); }
\foreach \x in {1,2,...,11} { \node [anchor=north] at (\x,-0.3) {\x}; }
\foreach \y in {1,2,...,11} { \node [anchor=east] at (-0.3,\y) {\y}; }
\node at (-0.5,-0.5) {0};
\end{tikzpicture}
\end{document}
For more details type: texdoc pgfmanual
from command prompt, or goto www.texdoc.net and search for pgfmanual. You can draw your graph entirely with tikz and/orpgfplots. If you find it difficult, draw the graph in geogebra, export it as tikz code and use the lines that produce the braces with appropriate co-ordinates (The lines are pointed to by comments in code).
Best Answer
Since it is your first question, and since we are supposed to be nice to new comers, here is a more verbose code for you.
You can actually draw the brackets unlike me. I have not used any particular position to put the ticks on the time line since I could not follow your pattern. This should serve as a starting point for you.