Q:

Unwanted Line in Tikz Graph

I have made a study guide for my students (we are studying rational functions) and when creating a key that involves graphs, I am getting a strange error. In my graph, there is an unwanted line at y=-1 but I don't see anything in my tex code that would create it. Can anyone explain to me what is going on?
While I'm here, I have 2 other questions to tidy up the graph 

How would I have tikz display every other label for a tick on the axis? For example, listing all of -9,...,9 creates a lot of clutter on the graph and listing -9,-7,...,9 would make it easier to read. 
Is there a neat way to create dashed asymptote lines? I can create them but they don't look neat at all.

\documentclass[11pt]{article}

\usepackage{amsfonts, colonequals, amsthm, amsmath, amssymb, mathrsfs, amscd, pgfplots, multicol, geometry}

\pgfplotsset{my style/.append style={axis x line=middle, axis y line=middle, xlabel={$x$}, ylabel={$y$}, axis equal }}

\geometry{textwidth=6.3in,textheight=9in}

\parindent=0in
\pagestyle{empty}

\begin{document}

Find \textbf{all} intercepts and asymptotes of \(F(x)=\frac{2x^2+16x-18}{x^2+x-6}\) and sketch a graph of \(F(x)\).\\

\(x\)-intercepts: \((-9,0), (1,0)\)\hspace{1cm} \(y\)-intercept: \((0,3)\)\hspace{1cm} VA: \(x=-3,2\)\hspace{1cm} HA: \(y=2\)

\begin{center}
\begin{tikzpicture}
\begin{axis}[my style, xtick={-9,-8,...,9}, ytick={-6,-5,...,6},xmin=-10, xmax=10,  ymin=-7, ymax=7]
\addplot[domain=-9.1:-3.1,samples=256] {(2*x^2+16*x-18)/(x^2+x-6)};
\addplot[domain=-2.9:1.9,samples=256] {(2*x^2+16*x-18)/(x^2+x-6)};
\addplot[domain=2.1:9.1,samples=256] {(2*x^2+16*x-18)/(x^2+x-6)};
\end{axis}
\end{tikzpicture}
\end{center}

\end{document}

A:

Two slightly different versions of the diagram are shown below. 
Regarding the asymptotes, a dashed line is a dashed line, it's not really clear to me in what sense the ones you made are "not neat". Anyways, you can draw a dashed line between the axis limits using
\draw [dashed] (axis cs:-3,\pgfkeysvalueof{/pgfplots/ymin}) -- 
               (axis cs:-3,\pgfkeysvalueof{/pgfplots/ymax});

where \pgfkeysvalueof{/pgfplots/ymin} is a way of accessing the lower limit of the axis, thus not having to explicitly write it. Similar for ymax, xmin, xmax. The coordinate "prefix" axis cs: is required because pgfplots doesn't use axis coordinates for \draw etc. unless you have compat=1.11 (or a newer version) in the \pgfplotsset in the preamble (see Why is tikzpicture destroyed when it is added inside the documentation). 
In the first of the diagrams below I added nodes/pins to the asymptotes, to indicate the values, mostly needed because I used a wider domain for the function. (Just a thought, which you can ignore as you see fit.) The number used for the pins, e.g. 200 in pin={200:$x=-3$}, is an angle, it indicates the direction of the pin.
For the tick locations, when you do a,b,...,c, the step used to fill in the rest of the series is defined by b-a, so as mentioned elsewhere, use xtick={-9,-7,...,9} to get x-ticks as you indicated. 
Other things: declare function is handy for cases where you need to use the same function several times, and if you want to use the same number of samples for all plots, it's more convenient to set it in the axis options. Never use \\ to end a paragraph. I added the parskip package instead of just setting the \parindent to zero. 
\documentclass[11pt]{article}
\usepackage{
  amsfonts,
  colonequals,
  amsthm,
  amsmath,
  amssymb,
  mathrsfs,
  amscd,
  pgfplots,
  multicol,
  geometry
}

\pgfplotsset{
  my style/.append style={
    axis x line=middle,
    axis y line=middle,
    xlabel={$x$},
    ylabel={$y$},
    axis equal
  }
}

\geometry{textwidth=6.3in,textheight=9in}

\usepackage{parskip} % better than just setting zero parindent

\pagestyle{empty}

\begin{document}

Find \emph{all} intercepts and asymptotes of \(F(x)=\frac{2x^2+16x-18}{x^2+x-6}\) and sketch a graph of \(F(x)\).

\(x\)-intercepts: \((-9,0), (1,0)\)\hfill
\(y\)-intercept: \((0,3)\) \hfill
VA: \(x=-3,2\)\hfill
HA: \(y=2\)

\begin{center}
\begin{tikzpicture}[
  declare function={F(\x)=(2*\x^2+16*\x-18)/(\x^2+\x-6);},
  every pin edge/.append style={solid} % otherwise the pin edges are dashed
]
\begin{axis}[
  my style,
  xmin=-15, xmax=15,
  ymin=-7, ymax=7,
  clip mode=individual,
  samples=50
]
\addplot[domain=-15:-3.1] {F(x)};
\addplot[domain=-2.9:1.9] {F(x)};
\addplot[domain=2.1:15] {F(x)};

\draw [dashed] (axis cs:-3,\pgfkeysvalueof{/pgfplots/ymin}) -- 
               (axis cs:-3,\pgfkeysvalueof{/pgfplots/ymax})
          coordinate[pos=0.9,pin={200:$x=-3$}](a);
\draw [dashed] (axis cs:2,\pgfkeysvalueof{/pgfplots/ymin}) -- 
               (axis cs:2,\pgfkeysvalueof{/pgfplots/ymax}) 
          coordinate[pos=0.9,pin={20:$x=2$}](a);

\draw [dashed] (axis cs:\pgfkeysvalueof{/pgfplots/xmin},2) -- 
               (axis cs:\pgfkeysvalueof{/pgfplots/xmax},2) node[right]{$y=2$};

\end{axis}
\end{tikzpicture}

\begin{tikzpicture}[
  declare function={F(\x)=(2*\x^2+16*\x-18)/(\x^2+\x-6);},
]
\begin{axis}[
  my style,
  xtick={-9,-7,...,9},
  ytick={-6,-4,...,6},
  xmin=-10, xmax=10,
  ymin=-7, ymax=7,
  clip mode=individual,
  samples=50
]
\addplot[domain=-9.1:-3.1] {F(x)};
\addplot[domain=-2.9:1.9] {F(x)};
\addplot[domain=2.1:9.1] {F(x)};

\draw [dashed] (axis cs:-3,\pgfkeysvalueof{/pgfplots/ymin}) -- 
               (axis cs:-3,\pgfkeysvalueof{/pgfplots/ymax});

\draw [dashed] (axis cs:2,\pgfkeysvalueof{/pgfplots/ymin}) -- 
               (axis cs:2,\pgfkeysvalueof{/pgfplots/ymax}); 

\draw [dashed] (axis cs:\pgfkeysvalueof{/pgfplots/xmin},2) -- 
               (axis cs:\pgfkeysvalueof{/pgfplots/xmax},2);

\end{axis}
\end{tikzpicture}
\end{center}

\end{document}

