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Cycloid
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<iframe 
    src="http://jsxgraph.uni-bayreuth.de/share/iframe/cycloid" 
    style="border: 1px solid black; overflow: hidden; width: 550px; aspect-ratio: 55 / 65;" 
    name="JSXGraph example: Cycloid" 
    allowfullscreen
></iframe>
This code has to 
<div id="board-0-wrapper" class="jxgbox-wrapper " style="width: 100%; ">
   <div id="board-0" class="jxgbox" style="aspect-ratio: 12 / 10; width: 100%;" data-ar="12 / 10"></div>
</div>

<script type = "text/javascript"> 
    /*
    This example is licensed under a 
    Creative Commons Attribution 4.0 International License.
    https://creativecommons.org/licenses/by/4.0/
    
    Please note you have to mention 
    The Center of Mobile Learning with Digital Technology
    in the credits.
    */
    
    const BOARDID = 'board-0';

    const board = JXG.JSXGraph.initBoard(BOARDID, {
        boundingbox: [-5, 5, 7, -5],
        axis: true
    });
     
    var r = board.create('slider', [[0.5, 4], [3.5, 4], [0, 1, 2]], {name: 'r'});
    var c = board.create('curve', [
        (t) => r.Value() * (t - Math.sin(t)),
        (t) => r.Value() * (1 - Math.cos(t)),
        -4 * Math.PI, 4* Math.PI
    ], {
        strokeWidth: 3
    });
 </script> 
/*
This example is licensed under a 
Creative Commons Attribution 4.0 International License.
https://creativecommons.org/licenses/by/4.0/

Please note you have to mention 
The Center of Mobile Learning with Digital Technology
in the credits.
*/

const BOARDID = 'your_div_id'; // Insert your id here!

const board = JXG.JSXGraph.initBoard(BOARDID, {
    boundingbox: [-5, 5, 7, -5],
    axis: true
});
 
var r = board.create('slider', [[0.5, 4], [3.5, 4], [0, 1, 2]], {name: 'r'});
var c = board.create('curve', [
    (t) => r.Value() * (t - Math.sin(t)),
    (t) => r.Value() * (1 - Math.cos(t)),
    -4 * Math.PI, 4* Math.PI
], {
    strokeWidth: 3
});
<jsxgraph width="100%" aspect-ratio="12 / 10" title="Cycloid" description="This construction was copied from JSXGraph examples database: BTW HERE SHOULD BE A GENERATED LINKuseGlobalJS="false">
   /*
   This example is licensed under a 
   Creative Commons Attribution 4.0 International License.
   https://creativecommons.org/licenses/by/4.0/
   
   Please note you have to mention 
   The Center of Mobile Learning with Digital Technology
   in the credits.
   */
   
   const board = JXG.JSXGraph.initBoard(BOARDID, {
       boundingbox: [-5, 5, 7, -5],
       axis: true
   });
    
   var r = board.create('slider', [[0.5, 4], [3.5, 4], [0, 1, 2]], {name: 'r'});
   var c = board.create('curve', [
       (t) => r.Value() * (t - Math.sin(t)),
       (t) => r.Value() * (1 - Math.cos(t)),
       -4 * Math.PI, 4* Math.PI
   ], {
       strokeWidth: 3
   });
</jsxgraph>

Cycloid

Curves
The cycloid through the origin, generated by a circle of radius r rolling over the $x$-axis on the positive side ($y \geq 0$), consists of the points $(x, y)$ with \[ \begin{aligned} x&=r(t-\sin t)\\ y&=r(1-\cos t). \end{aligned} \]
Web references 
// Define the id of your board in BOARDID

const board = JXG.JSXGraph.initBoard(BOARDID, {
    boundingbox: [-5, 5, 7, -5],
    axis: true
});
 
var r = board.create('slider', [[0.5, 4], [3.5, 4], [0, 1, 2]], {name: 'r'});
var c = board.create('curve', [
    (t) => r.Value() * (t - Math.sin(t)),
    (t) => r.Value() * (1 - Math.cos(t)),
    -4 * Math.PI, 4* Math.PI
], {
    strokeWidth: 3
});

license

This example is licensed under a Creative Commons Attribution 4.0 International License.
Please note you have to mention The Center of Mobile Learning with Digital Technology in the credits.