Graphing a rational function and a vertical line using pgfplots

Question

I would like the code that instructs TikZ to plot the graph for the rational function y=(x^{2}+5x+6)/(x^{2}+2x-3). I tried using addplot[very thin,blue]{frac(x^{2}+5x+6)/(x^{2}+2x-3)}. This graph has a vertical asymptote x = 1. What is the code for plotting this line - as a dashed line - with arrowheads?

\documentclass[10pt]{amsart}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.11}

\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}


\begin{document}

\hspace*{\fill}
\begin{tikzpicture}
\begin{axis}[axis equal image,
          xmax=8,ymax=7,
          axis lines=middle,
          restrict y to domain=-7:7,
          enlargelimits={abs=1cm},
          axis line style={latex-latex},
          ticklabel style={fill=white},
          ytick=\empty,
          xtick={-3}
          %xlabel=$x$,ylabel=$y$,
]
\addplot[domain=-10:10,mark=none,samples=10] {frac{x + 2}{x - 1} node [above left, yshift=3pt]{$\scriptstyle{y}=\frac{x^{2}+5x+6}{x^{2}+2x-3}$};
\draw [fill=white] (-3,0) circle [radius=1.5pt];
\end{axis}
\end{tikzpicture}
\hspace{\fill}

\end{document}

Answer 1

With my version of PGFplots, the \addplot command you provide doesn't even compile, so I had to guess what you want to achieve. In PGFplots you can just write fractions like this: 1/(x^2).

With Matthew's comment to Jake's answer to this question you can drawn an asymptote at x=1 by using the key vasymptote=1. In order to reliably truncate the plot without using many samples (orange curve), you can plot the two parts separately, specifying the domain so that is ends slightly before x=1 (black curve).

\documentclass[10pt]{amsart}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=1.8}

\pgfplotsset{vasymptote/.style={
    before end axis/.append code={
        \draw[densely dashed] ({rel axis cs:0,0} -| {axis cs:#1,0})
        -- ({rel axis cs:0,1} -| {axis cs:#1,0});
    }
}}

\begin{document}
\begin{tikzpicture}
\begin{axis}[
    axis equal image,
    axis lines=middle,
    xmin=-10,xmax=10,
    ymin=-10,ymax=10,
    enlargelimits={abs=1cm},
    axis line style={latex-latex},
    ticklabel style={fill=white},
    ytick={-10,10},
    xtick={-3,-10,10},
    vasymptote=1,
]
% This doesn't clip to y=-10:10 nicely
% because there are too few samples near the asymptote:
\addplot[very thick, orange, domain=-10:10,samples=200, restrict y to domain=-10:10]
    {(x^2+5*x+6)/(x^2+2*x-3)};

% Draw the two parts separately with individual domains:
\addplot[samples=50,domain=-10:1-0.27] {(x^2+5*x+6)/(x^2+2*x-3)};
\addplot[samples=50,domain=1+0.33:10]  {(x^2+5*x+6)/(x^2+2*x-3)};

\draw [fill=white] (-3,0) circle [radius=1.5pt]; % What is this?
\end{axis}
\end{tikzpicture}
\end{document}

Answer 2

Mark Wibrow's solution will produce a permanent catastrophic when plotpoints=14*(2^n)+1 (for any non-negative integer n).

So the following solution can be used to overcome such an issue (just for fun).

\documentclass[pstricks,border=20pt,12pt]{standalone}
\usepackage{pst-plot}
\def\f{(x+2)/(x-1)}

\begin{document}
\begin{psgraph}[algebraic,plotpoints=1000,Dy=5,Dx=2]{->}(0,0)(-11,-30)(11,35){20cm}{20cm}
    \psset{linecolor=red}
    \psplot{-10}{0.9}{\f}
    \psplot{1.1}{10}{\f}
    \psset{linestyle=dashed,linecolor=blue}
    \psline(1,-30)(1,35)
    \psline(-11,1)(11,1)
\end{psgraph}
\end{document}

Answer 3

Just for fun, with PSTricks :O

\documentclass[pstricks,border=5]{standalone}
\usepackage{pst-plot}
\begin{document}
\psset{xunit=20pt,yunit=20pt}
\begin{pspicture}*(-7.5,-7.5)(7.5,7.5)
  \psgrid[gridlabels=0,gridcolor=gray!25,subgridcolor=gray!10](-7,-7)(7,7)
  \psaxes[labels=none,ticksize=-2pt 2pt]{<->}(0,0)(-7,-7)(7,7)[$x$,-90][$y$,0]
  \psplot[linecolor=blue,plotstyle=line,algebraic, yMaxValue=7,plotpoints=1000]
      {-7}{7}{(x^2+5*x+6)/(x^2+2*x-3)} 
  \psline[linestyle=dashed](1,-7)(1,7)
  \rput(4.5,4){$\displaystyle f(x)=\frac{x^2+5x+6}{x^2+2x-3}$}
\end{pspicture}
\end{document} 

Answer 4

All you need are a couple of addplot commands:

  \addplot[domain=-10:10,samples=1000]
  {(x^2+5*x+6)/(x^2+2*x-3)};
  \addplot[dashed, purple, latex-latex]
  coordinates{(1,-6) (1,6)};

The first one should plot your real function, the second one draws a vertical line using the coordinates. the latex-latex option adds latex arrow heads to either end. If you don't want arrow heads on both ends, just remove the appropriate latex. If you want to the vertical line to be longer, just replace -6 and 6 in the coordinates with something bigger, eg. -15 and 15.

here's a full example:

\begin{tikzpicture}
\begin{axis}[axis equal image,
          xmin=-8, xmax=8,
          ymin=-7, ymax=7,
          axis lines=middle,
          restrict y to domain=-10:10,
          enlargelimits={abs=1cm},
          axis line style={-latex},
          ytick=\empty,
          xtick={-3},
          %xlabel=$x$,ylabel=$y$,
  ]
  \addplot[domain=-10:10,samples=1000]
  {(x^2+5*x+6)/(x^2+2*x-3)};
  \addplot[dashed, purple, latex-latex]
  coordinates{(1,-6) (1,6)};
\end{axis}
\end{tikzpicture}