\documentclass[]{standalone}
\usepackage{tikz}
\usepackage{pgfplots}
\usetikzlibrary{calc} 
\pgfplotsset{compat=1.14} 

\usepackage{xcolor}

%Draw Trochoids and Cycloid
%r fixed, b or Lambda varying
       
\tikzstyle{curves}=[
   smooth,samples=200,
   variable=\t,
   domain=0:4*pi,
]

%Radius of rolling circle
\pgfmathparse{1} \let\r\pgfmathresult
   
%Number of curves to draw.
\pgfmathparse{10} \let\N\pgfmathresult
   
%Relative position of rollong point. Lambda=\pm 1: cycloid
\pgfmathparse{-2.0} \let\LambdaMin\pgfmathresult
\pgfmathparse{2.0} \let\LambdaMax\pgfmathresult
%Steps of aLambda
\pgfmathparse{(\LambdaMax-\LambdaMin)/\N} \let\dLambda\pgfmathresult
  
%Variation of colors - color in [0,100].
\pgfmathparse{100/\N} \let\dcolor\pgfmathresult


\begin{document}
   \begin{tikzpicture}
       %Coordinate system
       \draw[-latex] (-\r,0) -- (4*pi+\r*\LambdaMax,0) node [right] {$x$};
       \draw[-latex] (0,-\r) -- (0,\r*\LambdaMax) node [above] {$y$};
              
       \foreach \n in {0,1,...,\N}
       {
           %\Lambda and color, \color
           \pgfmathparse{\LambdaMin+\n*\dLambda} \let\Lambda\pgfmathresult
           \pgfmathparse{\n*\dcolor} \let\color\pgfmathresult
          
           %cos and sin expects degrees: deg(\t)
           \draw
              [
                 curves,
                 %Convex combination of colors
                 color=orange!\color!cyan
              ]
              plot (
                 { \r*(  \t-\Lambda*sin(deg(\t))  ) },
                 { \r*(  1-\Lambda*cos(deg(\t)) }
              );
       }
       
       \draw
          [
             curves,thick,
             color=blue
          ]
          plot (  
             { \r*(  \t-sin(deg(\t))  ) },
             { \r*(  1-cos(deg(\t)) }
          );
      
       
       
   \end{tikzpicture}
\end{document}