Creating Rational Functions with Horizontal Asymptotes and More on Tikz

Question

I recently asked a question about how to use Tikz to create phase line diagrams. However I had a follow up question. The question is how I would go by creating a rational function or other types of functions during the plot portion of the code. The previous question is here - Is it possible to graph and draw phase lines in LaTex?

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What I initially wanted was to use Tikz to draw a graph plotting a horizontal asymptote at I=1 as a dashed line. I also wanted to draw two curves. One curve which approached I=1 from above, and another curve that approached I=1 from below. I looked everywhere for hours, but could not find what I needed to help me do this.

What I ended up doing was just messing around with the coding until I moved the two curves, and dashed line to look like I wanted it. I am hoping there is a better way to do this, as it took me a very long time.

\documentclass{amsart}
\usepackage{fullpage}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{graphicx}
\usepackage{caption}
\usepackage{epstopdf}
\usepackage{wrapfig}
\usepackage{pgfplots}
\usepackage{geometry}
\usepackage{amsmath}
\usepackage{pgfplots}
\pgfplotsset{compat=1.8}
\usepackage{mathtools}
\usepackage{tikz}

\newcommand*{\TickSize}{2pt}%

\newcommand*{\AxisMin}{0}%
\newcommand*{\AxisMax}{0}%

\newcommand*{\DrawHorizontalPhaseLine}[4][]{%
% #1 = axis tick labels
% #2 = right arrows positions as CSV
% #3 = left arrow positions as CSV
\gdef\AxisMin{0}%
\gdef\AxisMax{0}%
\edef\MyList{#2}% Allows for #1 to be both a macro or not
\foreach \X in \MyList {
    \draw  (\X,\TickSize) -- (\X,-\TickSize) node [below] {$\X$};
    \ifnum\AxisMin>\X
        \xdef\AxisMin{\X}%
    \fi
    \ifnum\AxisMax<\X
        \xdef\AxisMax{\X}%
    \fi
}

\edef\MyList{#3}% Allows for #2 to be both a macro or not
\foreach \X in \MyList {% Right arrows
    \draw [->] (\X-0.1,0) -- (\X,0);
    \ifnum\AxisMin>\X
        \xdef\AxisMin{\X}%
    \fi
    \ifnum\AxisMax<\X
        \xdef\AxisMax{\X}%
    \fi
}

\edef\MyList{#4}% Allows for #3 to be both a macro or not
\foreach \X in \MyList {% Left arrows
    \draw [<-] (\X-0.1,0) -- (\X,0);
    \ifnum\AxisMin>\X
        \xdef\AxisMin{\X}%
    \fi
    \ifnum\AxisMax<\X
        \xdef\AxisMax{\X}%
    \fi
}

\draw  (\AxisMin-1,0) -- (\AxisMax+1,0) node [right] {#1};
}%

\newcommand*{\DrawVerticalPhaseLine}[4][]{%
% #1 = axis tick labels
% #2 = up arrows positions as CSV
% #3 = down arrow positions as CSV
\gdef\AxisMin{0}%
\gdef\AxisMax{0}%
\edef\MyList{#2}% Allows for #1 to be both a macro or not
\foreach \X in \MyList {
    \draw  (-\TickSize,\X) -- (\TickSize,\X) node [right] {$\X$};
    \ifnum\AxisMin>\X
        \xdef\AxisMin{\X}%
    \fi
    \ifnum\AxisMax<\X
        \xdef\AxisMax{\X}%
    \fi
}

\edef\MyList{#3}% Allows for #2 to be both a macro or not
\foreach \X in \MyList {% Up arrows
    \draw [->] (0,\X-0.1) -- (0,\X);
    \ifnum\AxisMin>\X
        \xdef\AxisMin{\X}%
    \fi
    \ifnum\AxisMax<\X
        \xdef\AxisMax{\X}%
    \fi
}

\edef\MyList{#4}% Allows for #3 to be both a macro or not
\foreach \X in \MyList {% Down arrows
    \draw [<-] (0,\X+0.1) -- (0,\X);
    \ifnum\AxisMin>\X
        \xdef\AxisMin{\X}%
    \fi
    \ifnum\AxisMax<\X
        \xdef\AxisMax{\X}%
    \fi
}

\draw  (0,\AxisMin-1) -- (0,\AxisMax+1) node [above] {#1};
}%
\begin{document} 

\begin{tikzpicture}
  \draw[->] (0,0) -- (8,0) node[right] {$t$};
  \draw[->] (0,0) -- (0,5) node[above] {$I$};
   \draw[scale=0.5,domain=0:4,smooth,variable=\y,red]  plot ({\y*\y},{-   \y+8.1});  
   \draw[scale=0.5,domain=0:4,smooth,variable=\y,red]  plot ({\y*\y},{\y});  
  \draw[scale=0.5,domain=0:16,dashed,variable=\y,black] plot({\y},{4.045})node[right] {$I=1$};
 \end{tikzpicture}
  \end{document}

Answer 1

As this is a graph and not a picture, I would recommend you use pgfplots instead of just tike:

Code:

\documentclass[border=2pt]{standalone}
\usepackage{pgfplots}

%% http://tex.stackexchange.com/questions/17438/how-to-properly-scale-a-tikz-pgf-picture-which-has-a-beginaxis-endaxis
\pgfkeys{/pgfplots/Axis Labels At Tip/.style={
        xlabel style={
            at={(current axis.right of origin)}, 
            anchor=west,
        },
        ylabel style={
            at={(current axis.above origin)}, 
            yshift=1.5ex, anchor=center
        }
    }
}


\begin{document}
\begin{tikzpicture}
\begin{axis}[
    scale=0.5, 
    smooth, 
    xmax=16,
    ymax=8.5,
    axis x line=middle, 
    axis y line=center,
    ylabel=$I$,
    xlabel=$t$,
    axis on top=true,
    Axis Labels At Tip,
    clip=false,
    ]
  \addplot[domain=0:4, red] ({\x*\x},{-   \x+8.1});
  \addplot[domain=0:4, red] ({\x*\x},{\x});
  \addplot[domain=0:16, dashed, black] (\x,4.045) node[right] {$I=1$};
\end{axis}
\end{tikzpicture}
\end{document}