\documentclass[tikz]{standalone}

\usepackage{tikz}

\begin{document}
\usetikzlibrary{automata,positioning}

\iffalse
  $S = $ AACTACT
The suffix language of $S$ is $\{S, ACTACT, CTACT, TACT, ACT, CT, T\}$.

  \fi
\begin{tikzpicture}[shorten >=1pt,node distance=2cm,on grid, auto]
% TODO
  \node[state, initial] (q_0) {$q_{0}$};
  \node[state,right=of q_0] (q_1) {$q_{1}$};
  \node[state,right=of q_1] (q_2) {$q_{2}$};
  \node[state,right=of q_2] (q_3) {$q_{3}$};
   \node[state,right=of q_3] (q_4) {$q_{4}$};
  \node[state,right=of q_4] (q_5) {$q_{5}$};
  \node[state,right=of q_5] (q_6) {$q_{6}$};
  \node[state,accepting,right=of q_6] (q_7) {$q_{7}$};

  % Fix for CT
  \node[state,below=of q_2] (q_8) {$q_{8}$};
  \node[state,right=of q_8,accepting,yshift=-2cm] (q_9) {$q_{9}$};

  % Fix for T
% \node[state,above=of q_2,yshift=2cm,accepting] (q_10) {$q_{10}$};
  % S itself
  \path[->] (q_0) edge node {A} (q_1)
  (q_1) edge node {A} (q_2)
  (q_2) edge node {C} (q_3)
  (q_3) edge node {T} (q_4)
  (q_4) edge node {A} (q_5)
  (q_5) edge node {C} (q_6)
  (q_6) edge node {T} (q_7);

  % First suffix: ACTACT
  \path[->] (q_1) edge[bend left=30] node {C} (q_8)
  (q_9) edge[bend right=30] node {A} (q_5);

  % Second suffix: CTACT
  \path[->] (q_0) edge[bend right=30] node {C} (q_8)
  (q_8) edge node {T} (q_9);

  % Third suffix: TACT
 % \path[->] (q_0) edge[bend left=30] node {T} (q_10)
  %(q_10) edge[bend left=30] node {A} (q_5);


  % Fourth suffix: ACT
  % \path[->] (q_)

  % Fifth suffix: CT

  % Fix for T
  \path[->] (q_0) edge[bend right=40] node {T} (q_9);

\end{tikzpicture}

\end{document}