This is an attempt to collect some key methods on one page. Clearly, this discussion is not exhaustive, so I am hoping that this post gets complemented by others, who have other methods and/or opinions. (I made zero effort to precisely reproduce your curves, sorry.)
\documentclass{article}
\usepackage[margin=1in]{geometry}
\usepackage{multicol}
\usepackage{tikz}
\usetikzlibrary{hobby}
\begin{document}
\section*{A few ideas to draw smooth curves}
\begin{multicols}{2}
\subsection*{Your idea: use \texttt{out} and \texttt{in}}
This works, and the result looks smooth if you make sure that the \texttt{in}
of one point and the \texttt{out} of the next point differ by 180. In the
following pic, we have \texttt{\dots to[out=50,in=\textbf{210}] (1,2) circle(1pt)
to[\textbf{out=30},in=240] \dots} to illustrate this. Notice that the
\texttt{looseness} key can be of great help here.
\begin{tikzpicture}
\draw[latex-latex] (0,4) |- (6,0);
\draw[blue] (0,0) circle(1pt) to[out=50,in=210] (1,2) circle(1pt)
to[out=30,in=240,looseness=0.5] (3,3) circle(1pt);
\end{tikzpicture}
\subsection*{Your idea: find an analytic formula}
This is IMHO often the simplest way. One only needs to keep in mind a few basic
functions like $\sin$ and $\tanh$.
\begin{tikzpicture}
\draw[latex-latex] (0,4) |- (6,0);
\draw[blue] plot[variable=\x,domain=0:6,smooth]
({\x},{1+0.3*(tanh(3-\x)+1)*sin(180*\x)+0.2*\x});
\end{tikzpicture}
\subsection*{Use \texttt{plot[smooth] coordinates}}
A very convenient option, mentioned by Torj{\o}rn T.\ in the comments, is to use
\texttt{plot[smooth] coordinates}. You may want to play with the
\texttt{tension} key.
\begin{tikzpicture}
\draw[latex-latex] (0,4) |- (6,0);
\draw[blue] plot[smooth] coordinates
{(0,0) (1,2) (2,2) (3,3) (4,3)};
\draw[red] plot[smooth,tension=2] coordinates
{(0,0) (1,2) (2,2) (3,3) (4,3)};
\end{tikzpicture}
\subsection*{Use the \texttt{hobby} library}
The \texttt{hobby} library is sort of an el Dorado for smooth curve
constructors. It has way too many options to be listed here. One thing I find
very useful is the \texttt{tangent} style from the manual.
\begin{tikzpicture}[tangent/.style={%
in angle={(180+#1)},Hobby finish ,
designated Hobby path=next , out angle=#1,
}]
\draw[latex-latex] (0,4) |- (6,0);
\draw[blue] plot[smooth,hobby,tension=0.3] coordinates
{(0,0) (1,1.2) (2,2) (3,3) (4,3)};
\draw [red,use Hobby shortcut]
(0,0) .. ([tangent=30]1.5,1) .. ([tangent=-10]3,2) .. (5,1);
\end{tikzpicture}
I'd also like to mention that, when it comes to decorations, in my experience
the Hobby paths are advantageous. They often do not lead to \texttt{dimension too
large} errors when their almost identically looking \texttt{plot[smooth]}
counterparts do.
\subsection*{Not sure about \texttt{controls}}
There is also the possibility of using Bezier curves (without \texttt{hobby}).
Many users are able to use that to obtain great results. Unfortunately, I am not
one of those since IMHO the relation between position of the control points and
the outcome is not too obvious. But this is of course a very subjective
statement.
\end{multicols}
\end{document}
\draw plot[smooth] coordinates {<a suitable list of coordinates>};for the curves.(a1).. controls ++(45:1) and ++(30:0.5).. (a2)where the control points are relative polar coordinates. Somewhere on the site there is a style that will enable you to see the control vectors (i think it is calledshow curve controls), I use this method for most of my drawings when I'm trying to replicate or trace something.