Plot a function in latex (Do I have to use a formula?)

Question

This post is my help to beginners in LaTeX that are trying to plot functions and came out after trying to help with this post How to Draw 2 Functions In Tex, but my answer is more general and has to deal with basic tikz and pgf function plotting but also (and most) is a suggestion and a "how to" for plotting functions in case we really know the formula and in case we don't. The approach of the answer is to show to the newcomer that if he/she doesn't really know the actual formula, and even if he/she know a form, the math needed sometimes make the question out of this site's purposes and are closer to https://math.stackexchange.com/ but here.

Question: How can I plot a function like this:

Answer 1

Here is the code of a full answer that can be read as a pdf or directly:

\documentclass{article}
\usepackage{pgfplots}
\usepackage{amsmath}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}
\section{Ploting Mathematical Functions Using tikz or pgfplots}

Let's suppose we want to plot a function like this:

\begin{center}
\begin{tikzpicture}
  \draw[-,blue!30!green] (0,0)to[out=0,in=225](1.5,1);
  \draw[-,blue!30!green] (1.5,1)to[out=45,in=190](3,2);
\end{tikzpicture}
\end{center}

\subsection{The easy way (without real function):}

\subsubsection{Observe the function:}
We have just to pay attention to the properties of the function that has to be ploted. Here we need:

\begin{itemize}
\item The function has to start from about $(0,0)$ with almost zero (but possitive) derivative.
\item The function has to pass through $(1.5,1)$ with a derivative of about 1 (angle of $45^{\circ}$)
\item The function has to finish on point $(3,2)$ by a derivative of about 0 and possitive.
\end{itemize}

\subsubsection{tikz tools}

The tool we will use is the lines and arrows of tikz:

\begin{itemize}
\item An axis can be drawn with (here it is a x axis):
  \verb|\draw[->,thick] (0,0)--(3.5,0);|


\begin{itemize}
  \item In the above example (0,0) is the starting point and (3.5,0) is the finishing
  point.
\item \verb|->| means ``draw arrow'' for tikz.
\item \verb|--| connects these two points with an arrow (because of the previous)  
\end{itemize}

\item A grid can be ploted like this (here it is vertical lines between 0 and 3 by 0.5 units (of x) and between -0.2 and 2.5 (of y)):

\begin{verbatim}
\def\ymax{2.5}
\foreach \x in {0,0.5,...,3.5}{
\draw[-,thin] (\x,-0.2)--(\x,\ymax);}
\end{verbatim}

\begin{itemize}
\item In the above code\verb| \foreach| is a command that will be executed for each one of the values 0, 0.5, 1, 1.5 \ldots that are named (and can be used) as \verb|\x|
\item The line discribed before, but now it is a thin line.
\item In the finished example we will use two \verb|\foreach| commands to draw both $x$ and $y$ grids.
\end{itemize}


\item A curved line (here the function) can be drawn like this:

  \verb|\draw[-] (0,0) to[out=10,in=235](1.5,1);|
  \begin{itemize}
  \item (0,0) and (1.5,1) are the starting and ending points.
  \item \verb|out| is the angle which the line is going to follow to get out of the first point (See below).
    \item \verb|in| is the angle which the line is going to follow to get in the second point.
  \end{itemize}

  We can remember the angles by creating imaginatively an analog clock where
  3 o'clock is 0 degrees and 12 o'clock is 90 degrees etc:

  \begin{tikzpicture}
    \draw (2,-5) circle (3);
    \node at  (2,-5) {\LARGE clock};
    \node (A)at (5,-5) {$0^\circ$};
    \node (B)at (-1,-5) {$180^\circ$};
    \node (C)at (2,-2) {$90^\circ$};
    \node (D)at (2,-8) {$270^\circ$};
    \node[rotate=45] (E)at (4.1,-2.9) {$45^\circ$};
    \node at ([xshift=-1.cm]A) {3 o'clock=};
    \node at ([xshift=1.2cm]B) {=9 o'clock};
    \end{tikzpicture}


    A line that starts with $45^{\circ}$ by the point (0,0) and ends to a point (2,0) by $160^{\circ}$ can be drawn whith the command
    \verb|\draw (0,0)to[out=45,in=160](2,0);|:

    \begin{tikzpicture}
        \draw (0,0) circle (5pt);
        \draw (2,0) circle (5pt);
        \draw (0,0)to[out=45,in=160](2,0);
    \end{tikzpicture}

  \item A \verb|\node| is something that we will use here to add labels in the axes

    The node command can be used with coordinates and conetent, but the most usefull thing we have to know here is that can optionally have an `anchor' to be left,or east, or below etc of the point like:

    \verb|\node[left] at (3,2) {Content};|

    \end{itemize}

    \subsubsection{Plotting the function}

\begin{center}
  \begin{tikzpicture}[scale=2.5]
    \draw[->,thick] (-0.2,0)--(3.7,0);%x axis
    \draw[->,thick] (0,-0.2)--(0,2.7);%y axis
    \foreach \y in {0,0.5,...,2.5}{
      \draw[-,thin] (0.,\y)--(3.5,\y);
      \node[left] at (0,\y) {\y};
    }
    \foreach \x in {0,0.5,...,3.5}{
      \draw[-,thin] (\x,0)--(\x,2.5);
      \node[below] at (\x,0) {\x};
    }
  \draw[-,blue!30!green] (0,0)to[out=5,in=225](1.5,1);
  \draw[-,blue!30!green] (1.5,1)to[out=45,in=185](3,2);
\end{tikzpicture}
\end{center}


\subsection{The math way (using -extracting- the fanction):}

We want the graph to pass through $(0,0)$\footnote{Condition 1} and $(3,2)$\footnote{Condition 2} in a way that in $(1.5,1)$\footnote{Condition 3}
has a zero second derivative\footnote{Condition 4} and at $(3,2)$\footnote{Not a real condition} the first direvative will be almost zero.
(4 conditions)

The known shape comes from a function like:
$$f(x)=\frac{\alpha}{\beta+1\cdot e^{-\gamma (x+t)}}$$
(4 parameters)\footnote{The value of 1 as a mulriplier to $exp$ function
is just one of our free selected constand, since if we use any value, $\alpha$ and $\beta$ can always be written in a way that gives the same final function}

\begin{itemize}
\item Condition 1:\\
$$f(0)\approx 0 \Longrightarrow \alpha<< \beta+e^{-\gamma(t)} \Longrightarrow$$

\begin{equation}\alpha-\beta<< e^{-\gamma t}\label{eq:1}\end{equation}

\item Condition 4:\\
$$f^{\prime\prime}(1.5)=0 \Longrightarrow$$
But every derivative will give a product of $\alpha$, $exp(-\gamma(x+t))$ and $(x+t)$... And only $x-t$ can get zero, Thus, by knowing that $x=1.5$:
\begin{equation}
  t=-1.5
\end{equation}
\item Condition 2:\\
  $$f(3)=2\Longrightarrow$$
  \begin{equation}\label{eq:3}
    \alpha=2\beta+2\cdot e^{-1.5\gamma}
  \end{equation}
\item Condition 3:\\
  Finally condition 3 gives:
    $$f(1.5)=1\Longrightarrow$$
  \begin{equation}
    \alpha=\beta+1
  \end{equation}
  which combined with \eqref{eq:1} [if we want an errorof about 1\% units (the exp has to be about 100)in the last referred equation] gives:
  $\gamma\approx3$
  \begin{equation}
    \gamma=3
    \end{equation}
\end{itemize}


But Condition 2 (\eqref{eq:3})gives also:

$$\alpha=2\beta+2\cdot e^{-1.5\cdot3}\Longrightarrow$$
$$2\beta-\alpha=\beta-1=0.011\Longrightarrow$$
\begin{equation} \beta=1.011\end{equation}

  And Finaly we got the function:
\begin{equation}f(x)=\frac{2.011}{(1.011+e^{-3(x-1.5)})}\end{equation}






\begin{tikzpicture}
\begin{axis}[grid=both,xmax=3.25,ymax=2.5]
\addplot[blue,domain=0.1:3,samples=300]  {2.011/(1.011 + exp(-3*(x-1.5)))};
\end{axis}
\end{tikzpicture}


\end{document}

The above code generates this plot without using pgfplots or a formula, but with pure tikz curves:

And this after a mathematical analysis and a resulting formula:

Both codes (if you don't want to search all the above) are here:

\documentclass{article}
\usepackage{tikz}
\usepackage{pgfplots}

\begin{document}


\begin{center}
  \begin{tikzpicture}[scale=2.5]
    \draw[->,thick] (-0.2,0)--(3.7,0);%x axis
    \draw[->,thick] (0,-0.2)--(0,2.7);%y axis
    \foreach \y in {0,0.5,...,2.5}{
      \draw[-,thin] (0.,\y)--(3.5,\y);
      \node[left] at (0,\y) {\y};
    }
    \foreach \x in {0,0.5,...,3.5}{
      \draw[-,thin] (\x,0)--(\x,2.5);
      \node[below] at (\x,0) {\x};
    }
  \draw[-,blue!30!green] (0,0)to[out=5,in=225](1.5,1);
  \draw[-,blue!30!green] (1.5,1)to[out=45,in=185](3,2);
\end{tikzpicture}

\begin{tikzpicture}
\begin{axis}[grid=both,xmax=3.25,ymax=2.5]
\addplot[blue,domain=0.1:3,samples=300]  {2.011/(1.011 + exp(-3*(x-1.5)))};
\end{axis}
\end{tikzpicture}
\end{center}

\end{document}

Conclusion:

As you can see, the math needed, (even if the "type" of the formula is "almost known") are sometimes more complex that a simple latex user background and other times can be simple enough. So, if you really want to "extrtact" the actual formula... such questions are somehow offtopic and could be minegrated here: https://math.stackexchange.com/

But you (almost) always have the ability to use a number of curves like done above on first approach and get a result close to what you expect.

Answer 2

I know that you may not like this answer, but IMHO your procedure is a bit too complicated. So far I have always been able to produce reasonably close approximations by

• using elementary functions such as sin, cos, tan, exp etc.

• the to[in=...,out=...] syntax, which you also mention,

• adding some transformations (and yes, sometimes the polar transformation of the TikZ manual can be handy).

The function in your input looks awfully like a stretched (1-cos) function.

\documentclass[tikz,border=3.14pt]{standalone}
\begin{document}
\begin{tikzpicture}[>=latex]
    \draw (0,0) grid[step=0.5] (3.5,2.5);
    \draw [->] (0,0) --(3.7,0) node[below]{$x$};
    \draw [->] (0,0) --(0,2.7) node[left]{$y$};
    \draw[blue,thick] plot[variable=\x,domain=0:3] ({\x},{1-cos(deg(\x)*pi/3)});
\end{tikzpicture}
\end{document}

I am posting this merely to provide an alternative to your answer.

Answer 3

In case if we don't know the formula, one can use to[in=...,out=...]. Another possibility is to use Bezier curves.

\documentclass[12pt]{article}
\usepackage{tikz}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}





\pgfplotsset{cartesian/.style={%
            grid = both,
            minor tick num = 5,
            minor grid style = {line width=.1pt,draw=gray!10},
            axis lines = middle,
            axis line style={-stealth},
            xlabel style={below right},
            ylabel style={above left},
            xtick = {0,0.5,...,3.5},
            ytick = {0,0.5,...,2.5},
            xmin = 0,
            xmax =  3.5,
            ymin = 0,
            ymax =  2.5,
            width=1\linewidth,
            height=1\linewidth,
}}          

%===============================================================================
\begin{document}
\noindent
\begin{tikzpicture}
    \begin{axis}[cartesian]
        \draw[green!50!black] (0,0) to[out=0,in=185] (3,2);
    \end{axis}
\end{tikzpicture}


\end{document}

Answer 4

I'm usually just plotting a set of points as I found that an easier approach then trying to figure out a mathematical function which would end up right - or trying to draw bezier lines.

Now you might be wondering how to get the point data. For that I use the Engauge Digitizer tool (I'm just a user, not the creator). With some practice you can even use a hand scribble, scan it and generate points to plot.

If you are using this approach your compilation time might end up getting huge which is why I use an externalize approach for that, so the figures are being built as PDF only when the context changes and not every time.

I'm currently not able to test if I got everything in working order, but here is an outline of how it looks (maybe I can provide a better example later):

---

1. Generate a data file with Engauge Digitizer (in this example "elekguete.txt") looking somthing like this:

elekguete.txt:

0 1521.74
2 882.31
3 806.24
4 749.92
5 558.59
6 438.69
7 424.90
8 361.17
9 346.71
10 311.49
11 287.47
12 273.80
13 261.48
14 246.07
15 229.86
16 212.86
17 205.15
18 196.04

Then the file I'm using to generate the plot looks like this (of course to generate just a plot of the curve, you don't need the mathematical model):

\tikzsetnextfilename{elekguete}
\begin{tikzpicture}
        \begin{axis}[
        width = 0.7\textwidth,
        height = 0.35\textheight,
        grid=major,
        xmin = 0,
        xmax = 18,
        xlabel=Eintauchtiefe in \si{\mm},
        ylabel=elektrische Güte,
        legend style={anchor = north east, at={(0.98,0.98)}}
        ]

        \addplot[color=red,mark=x, only marks] table[x index=0,y index=1,header=false]
                {elekguete.txt};

        \addplot+[color=blue, no marks, domain=0:18]{204.23 +
        1290.7*exp(x/(-3.8778))};

        \end{axis}

\end{tikzpicture}

While using these settings, which are bloated now (I took it out of a document right now, like I said, I'll hopefully find time to produce a better, more minimal working example):

\usepackage{tikz}
\usetikzlibrary{positioning}
\usetikzlibrary{shapes}
\usetikzlibrary{shapes.geometric}
\usetikzlibrary{calc}
\usetikzlibrary{shapes.symbols}
\usetikzlibrary{shapes.multipart}
\usetikzlibrary{scopes}
\usetikzlibrary{decorations.shapes}
\usepackage{pgfplots}
\pgfplotsset{width=0.7\textwidth}
\pgfkeys{/pgf/number format/.cd,fixed,use comma,1000 sep={\,}}

\usetikzlibrary{external}
\tikzsetexternalprefix{externalize/}
\tikzset{external/system call={lualatex \tikzexternalcheckshellescape --source-specials -interaction=batchmode -jobname "\image" "\texsource"}}
\pgfkeys{/tikz/external/only named={true}}
\tikzexternalize