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</head>
<body>

  <div class="tree-container">
    
    <div class="root-node">
      <span>1. Evaluate</span>
      <code>f(−x)</code>
    </div>

    
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      <div class="branch-card branch-even active" id="card-even" onclick="selectBranch('even')">
        <div class="condition-tag">f(−x) = f(x)</div>
        <div class="arrow-down">↓</div>
        <div class="badge">EVEN</div>
        <div class="symmetry-label">y-axis<br>symmetry</div>
      </div>

      
      <div class="branch-card branch-odd" id="card-odd" onclick="selectBranch('odd')">
        <div class="condition-tag">f(−x) = −f(x)</div>
        <div class="arrow-down">↓</div>
        <div class="badge">ODD</div>
        <div class="symmetry-label">Origin<br>symmetry</div>
      </div>

      
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        <div class="condition-tag">Neither met</div>
        <div class="arrow-down">↓</div>
        <div class="badge">NEITHER</div>
        <div class="symmetry-label">No standard<br>symmetry</div>
      </div>
    </div>
  </div>

  
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    <div class="preview-info">
      <div class="preview-title" id="preview-title">Example: Even Function</div>
      <div class="preview-math" id="preview-math">f(x) = x² → f(−x) = (−x)² = x²</div>
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</html>

Identifying Even and Odd Symmetry in Functions

<!DOCTYPE html>
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<head>
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  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Even and Odd Functions Substitution Walkthrough</title>
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</head>
<body>

  <div class="stepper-header">
    <div class="step-pills">
      <button class="step-pill active" data-step="0">1. Replace (-x)</button>
      <button class="step-pill" data-step="1">2. Simplify</button>
      <button class="step-pill" data-step="2">3. Symmetry</button>
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      <button class="ctrl-btn" id="prevBtn" title="Previous" aria-label="Previous step">←</button>
      <button class="ctrl-btn" id="playBtn" title="Play/Pause" aria-label="Play/Pause">⏸</button>
      <button class="ctrl-btn" id="nextBtn" title="Next" aria-label="Next step">→</button>
    </div>
  </div>

  <div class="grid-wrap">
    
    <div class="func-col even-col">
      <div class="func-header">
        <span class="math-font">f(x) <span class="op">=</span> <span class="num">3</span>x<sup>4</sup> <span class="op">−</span> <span class="num">5</span>|x| <span class="op">+</span> <span class="num">2</span></span>
        <span class="col-badge">Test 1</span>
      </div>

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        <div class="step-label">Step 1: Input (-x)</div>
        <div class="step-math math-font">
          f(<span class="highlight-sub">−x</span>) <span class="op">=</span> <span class="num">3</span>(<span class="highlight-sub">−x</span>)<sup>4</sup> <span class="op">−</span> <span class="num">5</span>|<span class="highlight-sub">−x</span>| <span class="op">+</span> <span class="num">2</span>
        </div>
      </div>

      <div class="step-box" id="f-step2">
        <div class="step-label">Step 2: Simplify</div>
        <div class="step-math math-font">
          <span class="op">=</span> <span class="num">3</span>x<sup>4</sup> <span class="op">−</span> <span class="num">5</span>|x| <span class="op">+</span> <span class="num">2</span>
        </div>
        <div class="callouts" id="f-callouts" style="display:none;">
          <span class="bubble math-font">(−x)<sup>4</sup> <span class="op">=</span> x<sup>4</sup></span>
          <span class="bubble math-font">|−x| <span class="op">=</span> |x|</span>
        </div>
      </div>

      <div class="conclusion-box" id="f-conclusion">
        <span class="math-font">f(−x) <span class="op">=</span> f(x)</span>
        <span class="conclusion-badge">✦ EVEN</span>
      </div>
    </div>

    
    <div class="func-col odd-col">
      <div class="func-header">
        <span class="math-font">g(x) <span class="op">=</span> <span class="num">2</span>x<sup>3</sup> <span class="op">−</span> <span class="num">7</span>x</span>
        <span class="col-badge">Test 2</span>
      </div>

      <div class="step-box active-step" id="g-step1">
        <div class="step-label">Step 1: Input (-x)</div>
        <div class="step-math math-font">
          g(<span class="highlight-sub">−x</span>) <span class="op">=</span> <span class="num">2</span>(<span class="highlight-sub">−x</span>)<sup>3</sup> <span class="op">−</span> <span class="num">7</span>(<span class="highlight-sub">−x</span>)
        </div>
      </div>

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        <div class="step-label">Step 2: Simplify & Factor</div>
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        </div>
        <div class="callouts" id="g-callouts" style="display:none;">
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      </div>

      <div class="conclusion-box" id="g-conclusion">
        <span class="math-font">g(−x) <span class="op">=</span> −g(x)</span>
        <span class="conclusion-badge">✦ ODD</span>
      </div>
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      gConclusion.classList.toggle('highlight', idx === 2);
    }

    function nextStep() {
      showStep((currentStep + 1) % totalSteps);
    }

    function prevStep() {
      showStep((currentStep - 1 + totalSteps) % totalSteps);
    }

    function startAuto() {
      stopAuto();
      isPlaying = true;
      playBtn.textContent = '⏸';
      timer = setInterval(nextStep, 3500);
    }

    function stopAuto() {
      isPlaying = false;
      playBtn.textContent = '▶';
      if (timer) {
        clearInterval(timer);
        timer = null;
      }
    }

    function toggleAuto() {
      if (isPlaying) stopAuto();
      else startAuto();
    }

    pills.forEach((pill, i) => {
      pill.addEventListener('click', () => {
        stopAuto();
        showStep(i);
      });
    });

    prevBtn.addEventListener('click', () => { stopAuto(); prevStep(); });
    nextBtn.addEventListener('click', () => { stopAuto(); nextStep(); });
    playBtn.addEventListener('click', toggleAuto);

    showStep(0);
    startAuto();
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    .controls {
      display: flex;
      justify-content: center;
      align-items: center;
      gap: 12px;
    }

    .btn {
      background: #0284c7;
      color: #ffffff;
      border: none;
      border-radius: 6px;
      padding: 6px 14px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 6px;
      box-shadow: 0 1px 2px rgba(0,0,0,0.08);
      transition: background-color 0.2s ease, opacity 0.2s ease;
    }

    .btn:hover {
      background: #0369a1;
    }

    .btn:active {
      opacity: 0.85;
    }

    .graphs-container {
      display: flex;
      flex-direction: row;
      gap: 12px;
      justify-content: center;
      align-items: stretch;
      width: 100%;
    }

    .graph-card {
      flex: 1;
      min-width: 0;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 6px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 6px;
    }

    .header-row {
      display: flex;
      justify-content: space-between;
      align-items: center;
      width: 100%;
      padding: 0 4px;
    }

    .fn-title {
      font-size: 15px;
      font-weight: 700;
      font-family: "Cambria Math", "Times New Roman", serif;
      color: #0f172a;
    }

    .badge {
      font-size: 12px;
      font-weight: 700;
      padding: 2px 8px;
      border-radius: 12px;
    }

    .badge-odd {
      background: #dcfce7;
      color: #15803d;
      border: 1px solid #bbf7d0;
    }

    .badge-neither {
      background: #fee2e2;
      color: #b91c1c;
      border: 1px solid #fecaca;
    }

    .svg-wrap {
      width: 100%;
      max-width: 240px;
      aspect-ratio: 1 / 1;
    }

    svg {
      width: 100%;
      height: 100%;
      display: block;
      overflow: visible;
    }

    .rotatable {
      transform-origin: 100px 100px;
      transition: transform 0.8s cubic-bezier(0.4, 0, 0.2, 1);
    }

    .rotated .rotatable {
      transform: rotate(180deg);
    }

    .alert-box {
      background: #fffbeb;
      border: 1px solid #fde68a;
      border-radius: 6px;
      padding: 8px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #92400e;
      text-align: center;
    }

    @media (max-width: 480px) {
      body {
        padding: 8px;
        gap: 8px;
      }
      .graphs-container {
        gap: 8px;
      }
      .fn-title {
        font-size: 13px;
      }
      .badge {
        font-size: 11px;
        padding: 2px 6px;
      }
      .alert-box {
        font-size: 12px;
        padding: 6px 8px;
      }
    }
  </style>
</head>
<body>

  <div class="controls">
    <button class="btn" id="rotateBtn" onclick="toggleRotation()">
      <span id="btnIcon">↻</span> Rotate 180° around (0,0)
    </button>
  </div>

  <div class="graphs-container" id="graphsContainer">
    
    <div class="graph-card">
      <div class="header-row">
        <span class="fn-title">y = x³</span>
        <span class="badge badge-odd">Odd ✓</span>
      </div>
      <div class="svg-wrap">
        <svg viewBox="0 0 200 200">
          
          <path d="M 20 100 H 180 M 100 20 V 180" stroke="#94a3b8" stroke-width="1.5" />
          <path d="M 40 20 V 180 M 70 20 V 180 M 130 20 V 180 M 160 20 V 180 M 20 40 H 180 M 20 70 H 180 M 20 130 H 180 M 20 160 H 180" stroke="#e2e8f0" stroke-width="0.8" stroke-dasharray="2 2" />
          
          
          <polygon points="182,100 176,97 176,103" fill="#94a3b8" />
          <polygon points="100,18 97,24 103,24" fill="#94a3b8" />

          
          <path id="ghost1" stroke="#cbd5e1" stroke-width="2.5" stroke-dasharray="3 3" fill="none" />

          
          <g class="rotatable">
            <path id="curve1" stroke="#0284c7" stroke-width="2.5" stroke-linecap="round" fill="none" />
            
            <circle cx="100" cy="100" r="4.5" fill="#0284c7" />
          </g>

          
          <text x="106" y="114" font-size="11" font-weight="700" fill="#0284c7">(0, 0)</text>
        </svg>
      </div>
    </div>

    
    <div class="graph-card">
      <div class="header-row">
        <span class="fn-title">y = x³ + 5</span>
        <span class="badge badge-neither">Neither ✗</span>
      </div>
      <div class="svg-wrap">
        <svg viewBox="0 0 200 200">
          
          <path d="M 20 100 H 180 M 100 20 V 180" stroke="#94a3b8" stroke-width="1.5" />
          <path d="M 40 20 V 180 M 70 20 V 180 M 130 20 V 180 M 160 20 V 180 M 20 40 H 180 M 20 70 H 180 M 20 130 H 180 M 20 160 H 180" stroke="#e2e8f0" stroke-width="0.8" stroke-dasharray="2 2" />
          
          
          <polygon points="182,100 176,97 176,103" fill="#94a3b8" />
          <polygon points="100,18 97,24 103,24" fill="#94a3b8" />

          
          <line x1="100" y1="100" x2="100" y2="65" stroke="#f59e0b" stroke-width="1.5" stroke-dasharray="2 2" />
          <polygon points="100,60 97,66 103,66" fill="#f59e0b" />
          <text x="105" y="84" font-size="10" font-weight="700" fill="#d97706">+5</text>

          
          <path id="ghost2" stroke="#fca5a5" stroke-width="2.5" stroke-dasharray="3 3" fill="none" />

          
          <g class="rotatable">
            <path id="curve2" stroke="#7c3aed" stroke-width="2.5" stroke-linecap="round" fill="none" />
            
            <circle cx="100" cy="60" r="4.5" fill="#7c3aed" />
            <text x="106" y="58" font-size="11" font-weight="700" fill="#7c3aed">(0, 5)</text>
          </g>

          
          <circle cx="100" cy="100" r="3.5" fill="#64748b" />
          <text x="106" y="114" font-size="10" font-weight="600" fill="#64748b">(0, 0)</text>
        </svg>
      </div>
    </div>
  </div>

  <div class="alert-box">
    Odd symmetry requires rotation about (0,0) specifically, not (0,c).
  </div>

  <script>
    // Coordinate mapping:
    // x: [-3, 3] -> [10, 190], scaleX = 30 px/unit, center = 100
    // y: [-11, 11] -> [190, 10], scaleY = 8 px/unit, center = 100
    function mapX(x) { return 100 + x * 30; }
    function mapY(y) { return 100 - y * 8; }

    function generatePath(fn, xMin, xMax, step = 0.05) {
      let d = "";
      for (let x = xMin; x <= xMax + 0.001; x += step) {
        const y = fn(x);
        const px = mapX(x);
        const py = mapY(y);
        if (px >= 10 && px <= 190 && py >= 10 && py <= 190) {
          d += (d === "" ? "M " : " L ") + px.toFixed(1) + " " + py.toFixed(1);
        }
      }
      return d;
    }

    // Graph 1: y = x^3
    const path1 = generatePath(x => Math.pow(x, 3), -2.25, 2.25);
    document.getElementById('curve1').setAttribute('d', path1);
    document.getElementById('ghost1').setAttribute('d', path1);

    // Graph 2: y = x^3 + 5
    const path2 = generatePath(x => Math.pow(x, 3) + 5, -2.48, 1.83);
    document.getElementById('curve2').setAttribute('d', path2);
    document.getElementById('ghost2').setAttribute('d', path2);

    let isRotated = false;
    function toggleRotation() {
      isRotated = !isRotated;
      const container = document.getElementById('graphsContainer');
      const btn = document.getElementById('rotateBtn');
      if (isRotated) {
        container.classList.add('rotated');
        btn.innerHTML = '<span>↺</span> Reset to 0°';
      } else {
        container.classList.remove('rotated');
        btn.innerHTML = '<span>↻</span> Rotate 180° around (0,0)';
      }
    }
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: #ffffff;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }

    .main-wrap {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
    }

    .graph-container {
      position: relative;
      width: 100%;
      aspect-ratio: 1.15 / 1;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      overflow: hidden;
    }

    svg {
      width: 100%;
      height: 100%;
      display: block;
    }

    /* Interactive controls */
    .controls {
      display: flex;
      gap: 8px;
      width: 100%;
      justify-content: center;
    }

    .btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      border-radius: 6px;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 5px;
      transition: background-color 0.2s, border-color 0.2s;
    }
    .btn:active { background: #f1f5f9; }
    .btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
    }

    /* Cards below */
    .cards-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
      width: 100%;
    }

    .branch-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 10px;
      font-size: 13px;
      display: flex;
      flex-direction: column;
      gap: 3px;
      transition: box-shadow 0.3s, border-color 0.3s;
    }
    .branch-card.right { border-left: 4px solid #2563eb; }
    .branch-card.left { border-left: 4px solid #6366f1; }
    .branch-card.highlight {
      box-shadow: 0 0 0 2px #38bdf8;
      border-color: #38bdf8;
    }

    .card-title {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
    }
    .card-math {
      font-family: ui-monospace, monospace;
      font-weight: 600;
      font-size: 13px;
    }
    .card-math .c-blue { color: #2563eb; }
    .card-math .c-indigo { color: #6366f1; }

    /* Rotational line & points */
    #rotGroup {
      transform-origin: 150px 130px;
      transition: transform 1.2s cubic-bezier(0.4, 0, 0.2, 1);
    }
  </style>
</head>
<body>

<div class="main-wrap">
  
  <div class="graph-container">
    <svg viewBox="0 0 300 260">
      <defs>
        
        <pattern id="grid" width="25" height="25" patternUnits="userSpaceOnUse">
          <path d="M 25 0 L 0 0 0 25" fill="none" stroke="#e2e8f0" stroke-width="0.8" />
        </pattern>
      </defs>

      <rect width="300" height="260" fill="url(#grid)" />

      
      
      <line x1="15" y1="130" x2="285" y2="130" stroke="#94a3b8" stroke-width="1.5" />
      <line x1="150" y1="15" x2="150" y2="245" stroke="#94a3b8" stroke-width="1.5" />
      
      
      <path d="M 285 127 L 292 130 L 285 133 Z" fill="#94a3b8" />
      <path d="M 147 15 L 150 8 L 153 15 Z" fill="#94a3b8" />

      
      <text x="200" y="144" font-size="10" fill="#94a3b8" text-anchor="middle">2</text>
      <text x="100" y="144" font-size="10" fill="#94a3b8" text-anchor="middle">-2</text>
      <text x="142" y="45" font-size="10" fill="#94a3b8" text-anchor="end">4</text>
      <text x="142" y="222" font-size="10" fill="#94a3b8" text-anchor="end">-4</text>

      
      
      <path id="leftArm" d="M 150 130 Q 125 130 90 256" fill="none" stroke="#6366f1" stroke-width="3" stroke-linecap="round" />

      
      
      <path id="rightArm" d="M 150 130 Q 175 130 210 4" fill="none" stroke="#2563eb" stroke-width="3" stroke-linecap="round" />

      
      <g id="rotGroup">
        
        <line x1="100" y1="218" x2="200" y2="42" stroke="#f59e0b" stroke-width="2" stroke-dasharray="4 3" />

        
        <circle cx="200" cy="42" r="5" fill="#2563eb" stroke="#ffffff" stroke-width="2" />
        
        
        <circle cx="100" cy="218" r="5" fill="#6366f1" stroke="#ffffff" stroke-width="2" />

        
        <circle cx="150" cy="130" r="3.5" fill="#f59e0b" />
      </g>

      
      <text x="208" y="38" font-size="12" font-weight="700" fill="#2563eb">(2, 4)</text>
      <text x="92" y="232" font-size="12" font-weight="700" fill="#6366f1" text-anchor="end">(-2, -4)</text>
      <text x="156" y="125" font-size="11" font-weight="600" fill="#f59e0b">(0,0)</text>

      
      <path id="arcGuide" d="M 165 115 A 22 22 0 0 1 135 145" fill="none" stroke="#f59e0b" stroke-width="1.5" stroke-dasharray="2 2" />
      <polygon id="arcArrow" points="135,141 133,149 141,146" fill="#f59e0b" />
    </svg>
  </div>

  
  <div class="controls">
    <button class="btn active" id="btn0" onclick="setRotation(0)">Original (0°)</button>
    <button class="btn" id="btn180" onclick="setRotation(180)">Rotate 180° ⟲</button>
  </div>

  
  <div class="cards-grid">
    <div class="branch-card right" id="cardPos">
      <div class="card-title">x ≥ 0 branch</div>
      <div class="card-math"><span class="c-blue">p(2) = 2² = 4</span></div>
    </div>
    <div class="branch-card left" id="cardNeg">
      <div class="card-title">x &lt; 0 branch</div>
      <div class="card-math"><span class="c-indigo">p(-2) = -(-2)² = -4</span></div>
    </div>
  </div>
</div>

<script>
  let currentAngle = 0;

  function setRotation(deg) {
    currentAngle = deg;
    const group = document.getElementById('rotGroup');
    const btn0 = document.getElementById('btn0');
    const btn180 = document.getElementById('btn180');
    const cardPos = document.getElementById('cardPos');
    const cardNeg = document.getElementById('cardNeg');

    group.style.transform = `rotate(${deg}deg)`;

    if (deg === 180) {
      btn180.classList.add('active');
      btn0.classList.remove('active');
      cardNeg.classList.add('highlight');
      cardPos.classList.remove('highlight');
    } else {
      btn0.classList.add('active');
      btn180.classList.remove('active');
      cardPos.classList.add('highlight');
      cardNeg.classList.remove('highlight');
    }
  }
</script>

</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: #f8fafc;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }
    .cards-row {
      display: flex;
      flex-wrap: wrap;
      gap: 12px;
      width: 100%;
      max-width: 660px;
      justify-content: center;
    }
    .sym-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(30, 41, 69, 0.06);
      padding: 12px;
      flex: 1 1 290px;
      max-width: 320px;
      display: flex;
      flex-direction: column;
      align-items: center;
    }
    .card-title {
      font-size: 15px;
      font-weight: 700;
      color: #1e2945;
      margin-bottom: 2px;
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .card-formula {
      font-size: 13px;
      font-weight: 600;
      color: #2563eb;
      background: #eff6ff;
      padding: 2px 8px;
      border-radius: 6px;
      margin-bottom: 8px;
    }
    .card-formula.odd {
      color: #4f46e5;
      background: #eef2ff;
    }
    svg {
      width: 100%;
      height: auto;
      max-height: 220px;
      display: block;
      overflow: visible;
    }
    .axis { stroke: #cbd5e1; stroke-width: 1.5; }
    .grid-line { stroke: #f1f5f9; stroke-width: 1; }
    .curve-even { stroke: #2563eb; stroke-width: 2.5; fill: none; }
    .curve-odd { stroke: #4f46e5; stroke-width: 2.5; fill: none; }
    .mirror-axis { stroke: #f59e0b; stroke-width: 2; stroke-dasharray: 4 3; }
    .point-main { fill: #2563eb; }
    .point-sym { fill: #f59e0b; }
    .point-odd { fill: #4f46e5; }
    .coord-tag {
      font-size: 11px;
      font-weight: 600;
      fill: #1e2945;
      paint-order: stroke;
      stroke: #ffffff;
      stroke-width: 3px;
      stroke-linejoin: round;
    }
    .guide-line { stroke: #94a3b8; stroke-dasharray: 2 2; stroke-width: 1; }
    .action-arrow { stroke: #f59e0b; stroke-width: 1.75; fill: none; }
    .action-arrow-head { fill: #f59e0b; }
    .origin-dot { fill: #1e2945; }
    
    .controls {
      margin-top: 10px;
      display: flex;
      align-items: center;
      gap: 10px;
      width: 100%;
      max-width: 660px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      padding: 8px 14px;
      border-radius: 10px;
      font-size: 13px;
      font-weight: 600;
    }
    .slider-wrap {
      flex: 1;
      display: flex;
      align-items: center;
      gap: 8px;
    }
    input[type="range"] {
      flex: 1;
      accent-color: #2563eb;
      cursor: pointer;
    }
    .btn-animate {
      background: #2563eb;
      color: #fff;
      border: none;
      padding: 5px 12px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
      transition: background 0.2s;
    }
    .btn-animate:hover { background: #1d4ed8; }
  </style>
</head>
<body>

  <div class="cards-row">
    
    <div class="sym-card">
      <div class="card-title">Even Function</div>
      <div class="card-formula">f(-x) = f(x) · y-axis mirror</div>
      
      <svg id="svgEven" viewBox="0 0 260 210">
        
        <line x1="15" y1="175" x2="245" y2="175" class="axis" />
        <line x1="130" y1="15" x2="130" y2="195" class="mirror-axis" />
        
        <text x="248" y="179" font-size="11" fill="#94a3b8" font-weight="bold">x</text>
        <text x="134" y="24" font-size="11" fill="#f59e0b" font-weight="bold">y (mirror)</text>
        
        
        <path id="pathEven" class="curve-even" />
        
        
        <line id="evenGuide" class="guide-line" />
        <path id="evenTrans" class="action-arrow" />
        <polygon id="evenArrowR" class="action-arrow-head" />
        
        <circle id="ptEvenR" class="point-main" r="4.5" />
        <circle id="ptEvenL" class="point-sym" r="4.5" />
        
        <text id="lblEvenR" class="coord-tag" text-anchor="start">(x, y)</text>
        <text id="lblEvenL" class="coord-tag" text-anchor="end">(-x, y)</text>
      </svg>
    </div>

    
    <div class="sym-card">
      <div class="card-title">Odd Function</div>
      <div class="card-formula odd">f(-x) = -f(x) · 180° origin rotation</div>
      
      <svg id="svgOdd" viewBox="0 0 260 210">
        
        <line x1="15" y1="105" x2="245" y2="105" class="axis" />
        <line x1="130" y1="15" x2="130" y2="195" class="axis" />
        
        <text x="248" y="109" font-size="11" fill="#94a3b8" font-weight="bold">x</text>
        <text x="134" y="24" font-size="11" fill="#94a3b8" font-weight="bold">y</text>
        
        
        <circle cx="130" cy="105" r="3" class="origin-dot" />
        <text x="134" y="118" font-size="10" fill="#64748b" font-weight="600">(0,0)</text>

        
        <path id="pathOdd" class="curve-odd" />
        
        
        <line id="oddGuide" class="guide-line" />
        <path id="oddTrans" class="action-arrow" />
        <polygon id="oddArrowL" class="action-arrow-head" />

        <circle id="ptOddR" class="point-odd" r="4.5" />
        <circle id="ptOddL" class="point-sym" r="4.5" />
        
        <text id="lblOddR" class="coord-tag" text-anchor="start">(x, y)</text>
        <text id="lblOddL" class="coord-tag" text-anchor="end">(-x, -y)</text>
      </svg>
    </div>
  </div>

  <div class="controls">
    <div class="slider-wrap">
      <span>Point x:</span>
      <input type="range" id="xSlider" min="0.5" max="1.45" step="0.01" value="1.15">
      <span id="xVal" style="min-width:32px;">1.15</span>
    </div>
    <button class="btn-animate" id="btnAnim">▶ Trace</button>
  </div>

  <script>
    // Exact mathematical curve plots
    function buildEvenPath() {
      // Origin: (130, 175). Scale: 1 unit = 70px x, 70px y.
      let d = "";
      for (let x = -1.5; x <= 1.5; x += 0.05) {
        const px = 130 + x * 70;
        const py = 175 - (x * x) * 70;
        d += (d === "" ? "M" : "L") + px.toFixed(1) + "," + py.toFixed(1);
      }
      document.getElementById("pathEven").setAttribute("d", d);
    }

    function buildOddPath() {
      // Origin: (130, 105). Scale: 1 unit = 70px x, 25px y (so x^3 fits).
      let d = "";
      for (let x = -1.5; x <= 1.5; x += 0.05) {
        const px = 130 + x * 70;
        const py = 105 - (x * x * x) * 25;
        d += (d === "" ? "M" : "L") + px.toFixed(1) + "," + py.toFixed(1);
      }
      document.getElementById("pathOdd").setAttribute("d", d);
    }

    buildEvenPath();
    buildOddPath();

    function updatePoints(x) {
      document.getElementById("xVal").textContent = x.toFixed(2);

      // --- EVEN: y = x^2 ---
      const eOx = 130, eOy = 175;
      const ePx = eOx + x * 70;
      const ePy = eOy - (x * x) * 70;
      const eNx = eOx - x * 70;
      const eNy = ePy;

      const ptEvenR = document.getElementById("ptEvenR");
      const ptEvenL = document.getElementById("ptEvenL");
      const lblEvenR = document.getElementById("lblEvenR");
      const lblEvenL = document.getElementById("lblEvenL");
      const evenGuide = document.getElementById("evenGuide");
      const evenTrans = document.getElementById("evenTrans");
      const evenArrowR = document.getElementById("evenArrowR");

      ptEvenR.setAttribute("cx", ePx);
      ptEvenR.setAttribute("cy", ePy);
      ptEvenL.setAttribute("cx", eNx);
      ptEvenL.setAttribute("cy", eNy);

      lblEvenR.setAttribute("x", ePx + 7);
      lblEvenR.setAttribute("y", ePy - 4);
      lblEvenL.setAttribute("x", eNx - 7);
      lblEvenL.setAttribute("y", eNy - 4);

      evenGuide.setAttribute("x1", eNx);
      evenGuide.setAttribute("y1", eNy);
      evenGuide.setAttribute("x2", ePx);
      evenGuide.setAttribute("y2", ePy);

      // Reflection arc/arrow from right to left
      const arcOffset = 18;
      const arcY = Math.max(22, ePy - arcOffset);
      evenTrans.setAttribute("d", `M ${ePx} ${ePy} Q ${eOx} ${arcY - 10} ${eNx} ${eNy}`);
      
      // Arrowhead on left point
      evenArrowR.setAttribute("points", `${eNx},${eNy} ${eNx + 6},${eNy - 4} ${eNx + 4},${eNy + 4}`);

      // --- ODD: y = x^3 ---
      const oOx = 130, oOy = 105;
      const oPx = oOx + x * 70;
      const oPy = oOy - (x * x * x) * 25;
      const oNx = oOx - x * 70;
      const oNy = oOy + (x * x * x) * 25;

      const ptOddR = document.getElementById("ptOddR");
      const ptOddL = document.getElementById("ptOddL");
      const lblOddR = document.getElementById("lblOddR");
      const lblOddL = document.getElementById("lblOddL");
      const oddGuide = document.getElementById("oddGuide");
      const oddTrans = document.getElementById("oddTrans");
      const oddArrowL = document.getElementById("oddArrowL");

      ptOddR.setAttribute("cx", oPx);
      ptOddR.setAttribute("cy", oPy);
      ptOddL.setAttribute("cx", oNx);
      ptOddL.setAttribute("cy", oNy);

      lblOddR.setAttribute("x", oPx + 7);
      lblOddR.setAttribute("y", oPy - 4);
      lblOddL.setAttribute("x", oNx - 7);
      lblOddL.setAttribute("y", oNy + 12);

      oddGuide.setAttribute("x1", oNx);
      oddGuide.setAttribute("y1", oNy);
      oddGuide.setAttribute("x2", oPx);
      oddGuide.setAttribute("y2", oPy);

      // 180 degree rotation curve passing near/around the origin
      const dx = oPx - oOx;
      const dy = oPy - oOy;
      const radius = Math.hypot(dx, dy);
      oddTrans.setAttribute("d", `M ${oPx} ${oPy} A ${radius} ${radius} 0 0 1 ${oNx} ${oNy}`);

      // Arrowhead pointing towards (-x, -y) along rotation arc
      const angle = Math.atan2(oNy - oOy, oNx - oOx) - Math.PI / 2;
      const ax1 = oNx + 6 * Math.cos(angle - 0.5);
      const ay1 = oNy + 6 * Math.sin(angle - 0.5);
      const ax2 = oNx + 6 * Math.cos(angle + 0.5);
      const ay2 = oNy + 6 * Math.sin(angle + 0.5);
      oddArrowL.setAttribute("points", `${oNx},${oNy} ${ax1},${ay1} ${ax2},${ay2}`);
    }

    const slider = document.getElementById("xSlider");
    slider.addEventListener("input", (e) => {
      updatePoints(parseFloat(e.target.value));
    });

    // Auto-trace animation
    let animId = null;
    let animDir = 1;
    const btnAnim = document.getElementById("btnAnim");

    function animateStep() {
      let val = parseFloat(slider.value) + 0.008 * animDir;
      if (val >= 1.42) {
        val = 1.42;
        animDir = -1;
      } else if (val <= 0.6) {
        val = 0.6;
        animDir = 1;
      }
      slider.value = val;
      updatePoints(val);
      animId = requestAnimationFrame(animateStep);
    }

    btnAnim.addEventListener("click", () => {
      if (animId) {
        cancelAnimationFrame(animId);
        animId = null;
        btnAnim.innerHTML = "▶ Trace";
      } else {
        btnAnim.innerHTML = "⏸ Pause";
        animId = requestAnimationFrame(animateStep);
      }
    });

    // Initial render
    updatePoints(1.15);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      height: 100%;
      padding: 12px;
      user-select: none;
    }

    .header-bar {
      display: flex;
      flex-wrap: wrap;
      justify-content: space-between;
      align-items: center;
      width: 100%;
      max-width: 480px;
      gap: 8px;
    }

    .math-pill {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 4px 10px;
      font-size: 14px;
      font-weight: 600;
      display: flex;
      align-items: center;
      gap: 6px;
    }

    .badge {
      font-size: 12px;
      font-weight: 700;
      padding: 4px 10px;
      border-radius: 20px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }
    .badge-up { background: #dbeafe; color: #1d4ed8; }
    .badge-down { background: #fef3c7; color: #b45309; }
    .badge-flat { background: #f1f5f9; color: #64748b; }

    .graph-container {
      width: 100%;
      max-width: 480px;
      flex: 1;
      min-height: 220px;
      position: relative;
    }

    svg {
      width: 100%;
      height: 100%;
      display: block;
      overflow: visible;
    }

    .controls {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .slider-row {
      display: flex;
      align-items: center;
      gap: 12px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      padding: 8px 14px;
      border-radius: 10px;
    }

    .slider-row label {
      font-weight: 700;
      font-size: 15px;
      min-width: 55px;
      color: #0f172a;
    }

    input[type=range] {
      flex: 1;
      height: 6px;
      border-radius: 3px;
      background: #cbd5e1;
      outline: none;
      -webkit-appearance: none;
      cursor: pointer;
    }
    input[type=range]::-webkit-slider-thumb {
      -webkit-appearance: none;
      width: 20px;
      height: 20px;
      border-radius: 50%;
      background: #2563eb;
      cursor: pointer;
      box-shadow: 0 1px 4px rgba(0,0,0,0.25);
    }

    .presets {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 6px;
    }

    .preset-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      padding: 6px 4px;
      font-size: 13px;
      font-weight: 600;
      color: #334155;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s ease;
    }
    .preset-btn.active {
      background: #2563eb;
      border-color: #2563eb;
      color: #ffffff;
      box-shadow: 0 2px 4px rgba(37,99,235,0.2);
    }
  </style>
</head>
<body>

  <div class="header-bar">
    <div class="math-pill" id="funcExpr">f(x) = x^r</div>
    <div class="math-pill" id="derivExpr">f''(1) = r(r-1)</div>
    <div class="badge badge-up" id="concavityBadge">Concave Up</div>
  </div>

  <div class="graph-container">
    <svg id="plotSvg" viewBox="0 0 360 260">
      <defs>
        <clipPath id="graphClip">
          <rect x="40" y="10" width="300" height="220" />
        </clipPath>
      </defs>

      
      <g stroke="#f1f5f9" stroke-width="1.5">
        <line x1="40" y1="170" x2="340" y2="170" />
        <line x1="40" y1="110" x2="340" y2="110" />
        <line x1="40" y1="50" x2="340" y2="50" />
        <line x1="140" y1="10" x2="140" y2="230" />
        <line x1="240" y1="10" x2="240" y2="230" />
      </g>

      
      <g stroke="#64748b" stroke-width="1.5">
        <line x1="40" y1="230" x2="345" y2="230" />
        <line x1="40" y1="230" x2="40" y2="5" />
        
        <polygon points="345,227 352,230 345,233" fill="#64748b" />
        <polygon points="37,5 40,-2 43,5" fill="#64748b" />
      </g>

      
      <g fill="#94a3b8" font-size="11" font-weight="600" text-anchor="middle">
        <text x="40" y="244">0</text>
        <text x="140" y="244">1</text>
        <text x="240" y="244">2</text>
        <text x="340" y="244">3</text>
        <text x="348" y="222" fill="#64748b">x</text>

        <text x="28" y="174" text-anchor="end">1</text>
        <text x="28" y="114" text-anchor="end">2</text>
        <text x="28" y="54" text-anchor="end">3</text>
        <text x="32" y="12" fill="#64748b">y</text>
      </g>

      
      <line x1="140" y1="230" x2="140" y2="170" stroke="#cbd5e1" stroke-dasharray="3,3" stroke-width="1.2" />
      <line x1="40" y1="170" x2="140" y2="170" stroke="#cbd5e1" stroke-dasharray="3,3" stroke-width="1.2" />

      
      <path id="curvePath" clip-path="url(#graphClip)" fill="none" stroke="#2563eb" stroke-width="3.5" stroke-linecap="round" stroke-linejoin="round" />

      
      <circle cx="140" cy="170" r="5" fill="#2563eb" stroke="#ffffff" stroke-width="2" />
      <text x="146" y="164" fill="#1e2945" font-size="11" font-weight="700">(1, 1)</text>
    </svg>
  </div>

  <div class="controls">
    <div class="slider-row">
      <label id="rValLabel">r = 1.33</label>
      <input type="range" id="rSlider" min="-1.5" max="2.5" step="0.05" value="1.33">
    </div>

    <div class="presets">
      <button class="preset-btn" data-r="-0.50">r &lt; 0<br><span style="font-size:11px;opacity:0.85">x⁻⁰·⁵</span></button>
      <button class="preset-btn" data-r="0.67">0 &lt; r &lt; 1<br><span style="font-size:11px;opacity:0.85">x²/³</span></button>
      <button class="preset-btn active" data-r="1.33">r &gt; 1<br><span style="font-size:11px;opacity:0.85">x⁴/³</span></button>
    </div>
  </div>

  <script>
    const ORIGIN_X = 40;
    const ORIGIN_Y = 230;
    const SCALE = 100; // 1 unit = 100px

    const slider = document.getElementById('rSlider');
    const rValLabel = document.getElementById('rValLabel');
    const curvePath = document.getElementById('curvePath');
    const funcExpr = document.getElementById('funcExpr');
    const derivExpr = document.getElementById('derivExpr');
    const concavityBadge = document.getElementById('concavityBadge');
    const presetBtns = document.querySelectorAll('.preset-btn');

    function renderMath(element, tex, fallback) {
      if (window.katex) {
        try {
          katex.render(tex, element, { throwOnError: false });
          return;
        } catch(e) {}
      }
      element.innerHTML = fallback;
    }

    function updatePlot(r) {
      rValLabel.textContent = `r = ${r.toFixed(2)}`;

      // Calculate path points
      const points = [];
      const xMin = (r < 0) ? 0.12 : 0.0;
      const xMax = 3.1;
      const steps = 140;

      for (let i = 0; i <= steps; i++) {
        const x = xMin + (i / steps) * (xMax - xMin);
        const y = Math.pow(x, r);

        if (Number.isFinite(y)) {
          const svgX = ORIGIN_X + x * SCALE;
          const svgY = ORIGIN_Y - y * SCALE;
          // Only add points within reasonable vertical bounds to avoid clipping artifacts
          if (svgY >= -20 && svgY <= 260) {
            points.push(`${svgX.toFixed(1)},${svgY.toFixed(1)}`);
          }
        }
      }

      if (points.length > 1) {
        curvePath.setAttribute('d', 'M ' + points.join(' L '));
      }

      // Concavity at x = 1: f''(1) = r(r - 1)
      const d2 = r * (r - 1);
      const signStr = d2 > 0.001 ? '+' : '';
      const d2Formatted = Math.abs(d2) < 0.001 ? '0.00' : signStr + d2.toFixed(2);

      // Render math labels
      const rStr = r.toFixed(2);
      renderMath(funcExpr, `f(x) = x^{${rStr}}`, `f(x) = x<sup>${rStr}</sup>`);
      renderMath(derivExpr, `f''(1) = ${d2Formatted}`, `f''(1) = ${d2Formatted}`);

      // Concavity state badge & styling
      if (d2 > 0.02) {
        concavityBadge.textContent = "Concave Up (f'' > 0)";
        concavityBadge.className = "badge badge-up";
        curvePath.setAttribute('stroke', '#2563eb');
      } else if (d2 < -0.02) {
        concavityBadge.textContent = "Concave Down (f'' < 0)";
        concavityBadge.className = "badge badge-down";
        curvePath.setAttribute('stroke', '#d97706');
      } else {
        concavityBadge.textContent = "Linear / Flat (f'' = 0)";
        concavityBadge.className = "badge badge-flat";
        curvePath.setAttribute('stroke', '#64748b');
      }

      // Update preset buttons active state
      presetBtns.forEach(btn => {
        const btnR = parseFloat(btn.dataset.r);
        const isMatch = Math.abs(btnR - r) < 0.15;
        btn.classList.toggle('active', isMatch);
      });
    }

    slider.addEventListener('input', (e) => {
      updatePlot(parseFloat(e.target.value));
    });

    presetBtns.forEach(btn => {
      btn.addEventListener('click', () => {
        const val = parseFloat(btn.dataset.r);
        slider.value = val;
        updatePlot(val);
      });
    });

    // Initial render
    updatePlot(parseFloat(slider.value));
  </script>
</body>
</html>

Analyzing Power Functions and Fractional Exponents

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; user-select: none; -webkit-user-select: none; }
    body {
      width: 100%;
      font-family: system-ui, -apple-system, sans-serif;
      color: #1e2945;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      padding: 8px 4px;
    }
    .toolbar {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 6px;
      margin-bottom: 8px;
      width: 100%;
      max-width: 440px;
    }
    .btn-pin {
      font-size: 12px;
      font-weight: 600;
      padding: 4px 10px;
      border-radius: 14px;
      border: 1px solid #cbd5e1;
      background: #f8fafc;
      color: #475569;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
      transition: background 0.2s, color 0.2s, border-color 0.2s;
    }
    .btn-pin.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
    }
    .graph-container {
      position: relative;
      width: 100%;
      max-width: 440px;
      aspect-ratio: 1 / 0.85;
      touch-action: none;
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
      overflow: visible;
    }
    .callout {
      position: absolute;
      background: rgba(255, 255, 255, 0.95);
      border: 1px solid #94a3b8;
      border-radius: 6px;
      padding: 4px 8px;
      font-size: 11px;
      font-weight: 600;
      color: #0f172a;
      pointer-events: none;
      box-shadow: 0 2px 6px rgba(0,0,0,0.08);
      transform: translate(-50%, -120%);
      white-space: nowrap;
      display: none;
    }
    .callout.visible {
      display: block;
    }
    .readout {
      margin-top: 6px;
      font-size: 13px;
      font-weight: 600;
      color: #334155;
      font-family: ui-monospace, monospace;
      background: #f1f5f9;
      padding: 4px 12px;
      border-radius: 6px;
      border: 1px solid #e2e8f0;
      display: flex;
      gap: 16px;
    }
  </style>
</head>
<body>

  <div class="toolbar">
    <button class="btn-pin active" id="btnCusp" onclick="togglePin('cusp')">📍 Cusp</button>
    <button class="btn-pin" id="btnConcave" onclick="togglePin('concave')">📍 Concavity</button>
    <button class="btn-pin" id="btnSymmetry" onclick="togglePin('symmetry')">📍 Symmetry</button>
  </div>

  <div class="graph-container" id="container">
    <svg id="svgGraph" viewBox="-4 -0.6 8 3.8">
      <defs>
        <marker id="arrow" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="4" markerHeight="4" orient="auto-start-reverse">
          <path d="M 0 1.5 L 8 5 L 0 8.5 z" fill="#94a3b8" />
        </marker>
      </defs>

      
      <line x1="-3.8" y1="0" x2="3.8" y2="0" stroke="#94a3b8" stroke-width="0.04" marker-end="url(#arrow)" />
      <line x1="0" y1="-0.4" x2="0" y2="3.0" stroke="#94a3b8" stroke-width="0.04" marker-end="url(#arrow)" />

      
      <g stroke="#cbd5e1" stroke-width="0.02">
        <line x1="-3" y1="-0.08" x2="-3" y2="0.08" />
        <line x1="-2" y1="-0.08" x2="-2" y2="0.08" />
        <line x1="-1" y1="-0.08" x2="-1" y2="0.08" />
        <line x1="1" y1="-0.08" x2="1" y2="0.08" />
        <line x1="2" y1="-0.08" x2="2" y2="0.08" />
        <line x1="3" y1="-0.08" x2="3" y2="0.08" />
        <line x1="-0.08" y1="1" x2="0.08" y2="1" />
        <line x1="-0.08" y1="2" x2="0.08" y2="2" />
      </g>

      
      <text x="3.7" y="-0.12" font-size="0.25" fill="#64748b" text-anchor="end">x</text>
      <text x="0.2" y="2.95" font-size="0.25" fill="#64748b">y</text>
      <text x="1" y="-0.15" font-size="0.2" fill="#94a3b8" text-anchor="middle">1</text>
      <text x="-1" y="-0.15" font-size="0.2" fill="#94a3b8" text-anchor="middle">-1</text>
      <text x="-0.2" y="1.06" font-size="0.2" fill="#94a3b8" text-anchor="end">1</text>

      
      <g id="symHelper" opacity="0" style="transition: opacity 0.3s ease;">
        <line x1="-2" y1="1.587" x2="2" y2="1.587" stroke="#8b5cf6" stroke-width="0.03" stroke-dasharray="0.1, 0.08" />
        <circle cx="-2" cy="1.587" r="0.07" fill="#8b5cf6" />
        <circle cx="2" cy="1.587" r="0.07" fill="#8b5cf6" />
      </g>

      
      <g id="concHelper" opacity="0" style="transition: opacity 0.3s ease;">
        <path d="M 1.2 1.45 Q 2.0 1.95 2.8 2.25" fill="none" stroke="#f59e0b" stroke-width="0.05" stroke-linecap="round" />
      </g>

      
      <path id="curvePath" fill="none" stroke="#0284c7" stroke-width="0.07" stroke-linejoin="round" stroke-linecap="round" />

      
      <line id="tangentLine" stroke="#ef4444" stroke-width="0.035" stroke-dasharray="0.08, 0.06" />

      
      <circle id="probePoint" cx="1" cy="1" r="0.11" fill="#0284c7" stroke="#ffffff" stroke-width="0.03" style="cursor: grab;" />
    </svg>

    
    <div id="calloutCusp" class="callout visible" style="left: 50%; top: 78%;">
      Sharp Cusp (0,0)<br>Slope → ±∞
    </div>
    <div id="calloutConcave" class="callout" style="left: 74%; top: 38%;">
      Concave Down<br>f''(x) &lt; 0
    </div>
    <div id="calloutSymmetry" class="callout" style="left: 50%; top: 32%;">
      Even: f(-x) = f(x)
    </div>
  </div>

  <div class="readout">
    <span id="coordDisplay">x: 1.00, y: 1.00</span>
    <span id="slopeDisplay">f'(x): 0.67</span>
  </div>

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      currentX = Math.max(-3.4, Math.min(3.4, x));
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      const y = Math.pow(absX, 2/3);

      probePoint.setAttribute('cx', currentX);
      probePoint.setAttribute('cy', y);

      // Tangent line & slope calculation: f'(x) = (2/3) * sign(x) * |x|^(-1/3)
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        const dx = 0.6 / Math.sqrt(1 + slope * slope);
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      coordDisplay.textContent = `x: ${currentX.toFixed(2)}, y: ${y.toFixed(2)}`;
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      const clientX = e.touches ? e.touches[0].clientX : e.clientX;
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    }

    container.addEventListener('pointerdown', (e) => {
      isDragging = true;
      updateProbe(getXFromEvent(e));
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    window.addEventListener('pointermove', (e) => {
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      updateProbe(getXFromEvent(e));
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    window.addEventListener('pointerup', () => { isDragging = false; });
    window.addEventListener('pointercancel', () => { isDragging = false; });

    // Pins state
    const pins = { cusp: true, concave: false, symmetry: false };

    function togglePin(type) {
      pins[type] = !pins[type];
      
      const btnMap = { cusp: 'btnCusp', concave: 'btnConcave', symmetry: 'btnSymmetry' };
      const calloutMap = { cusp: 'calloutCusp', concave: 'calloutConcave', symmetry: 'calloutSymmetry' };

      document.getElementById(btnMap[type]).classList.toggle('active', pins[type]);
      document.getElementById(calloutMap[type]).classList.toggle('visible', pins[type]);

      if (type === 'symmetry') {
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        document.getElementById('concHelper').style.opacity = pins.concave ? '1' : '0';
      }
    }

    // Flip SVG y-axis to match standard mathematical Cartesian coordinates
    document.getElementById('svgGraph').style.transform = 'scaleY(-1)';

    buildPath();
    updateProbe(1.2);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      -webkit-font-smoothing: antialiased;
    }

    body {
      background-color: #f8fafc;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #0f172a;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      width: 100%;
    }

    /* Top Math Header */
    .header-bar {
      display: flex;
      align-items: center;
      justify-content: space-between;
      flex-wrap: wrap;
      gap: 8px;
      padding-bottom: 4px;
      border-bottom: 1px solid #e2e8f0;
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    .formula-display {
      font-size: 20px;
      font-weight: 700;
      font-family: "Cambria Math", "KaTeX_Main", "Times New Roman", serif;
      letter-spacing: 0.5px;
      display: flex;
      align-items: center;
      gap: 6px;
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    .token-p {
      color: #0284c7;
      font-weight: 800;
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    .token-q {
      color: #7c3aed;
      font-weight: 800;
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    .token-legend {
      display: flex;
      align-items: center;
      gap: 10px;
      font-size: 11px;
      font-weight: 600;
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      display: inline-flex;
      align-items: center;
      gap: 4px;
      padding: 2px 6px;
      border-radius: 4px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
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    .legend-p { color: #0284c7; }
    .legend-q { color: #7c3aed; }

    /* Preset Toolbar */
    .tabs-bar {
      display: flex;
      flex-wrap: wrap;
      gap: 6px;
    }

    .tab-btn {
      flex: 1 1 calc(25% - 6px);
      min-width: 70px;
      padding: 6px 4px;
      border-radius: 6px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: #334155;
      font-size: 12px;
      font-weight: 600;
      cursor: pointer;
      text-align: center;
      transition: background-color 0.2s ease, color 0.2s ease, border-color 0.2s ease;
      font-family: inherit;
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    .tab-btn:hover {
      background: #f1f5f9;
    }

    .tab-btn.active {
      background: #0284c7;
      border-color: #0284c7;
      color: #ffffff;
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    /* Main Split */
    .content-grid {
      display: grid;
      grid-template-columns: 1fr 1.35fr;
      gap: 10px;
      align-items: stretch;
    }

    @media (max-width: 480px) {
      .content-grid {
        grid-template-columns: 1fr;
      }
      .tab-btn {
        flex: 1 1 calc(50% - 6px);
      }
    }

    /* Left Panel */
    .info-panel {
      display: flex;
      flex-direction: column;
      gap: 8px;
      justify-content: space-between;
    }

    .info-tile {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 10px;
      display: flex;
      flex-direction: column;
      gap: 3px;
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    .tile-label {
      font-size: 10px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
    }

    .radical-text {
      font-size: 18px;
      font-family: "Cambria Math", "KaTeX_Main", "Times New Roman", serif;
      font-weight: 700;
      color: #0f172a;
    }

    .badge {
      display: inline-block;
      font-size: 11px;
      font-weight: 700;
      padding: 3px 8px;
      border-radius: 5px;
      width: fit-content;
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    .badge-teal {
      background: #f0fdf4;
      color: #166534;
      border: 1px solid #bbf7d0;
    }

    .badge-amber {
      background: #fffbeb;
      color: #92400e;
      border: 1px solid #fde68a;
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    .rule-box {
      font-size: 11px;
      line-height: 1.35;
      color: #334155;
      background: #f8fafc;
      border-radius: 5px;
      padding: 6px;
      border-left: 3px solid #0284c7;
    }

    /* Right Panel / Graph */
    .graph-container {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      position: relative;
      width: 100%;
      aspect-ratio: 5 / 4;
      overflow: hidden;
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    svg {
      width: 100%;
      height: 100%;
      display: block;
    }

    .axis-line {
      stroke: #64748b;
      stroke-width: 1.5;
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    .grid-line {
      stroke: #f1f5f9;
      stroke-width: 1;
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    .tick-label {
      font-size: 9px;
      fill: #94a3b8;
      font-family: system-ui, sans-serif;
      text-anchor: middle;
      user-select: none;
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    .curve-path {
      fill: none;
      stroke: #0284c7;
      stroke-width: 3.2;
      stroke-linecap: round;
      stroke-linejoin: round;
      transition: d 0.25s ease;
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    .forbidden-zone {
      fill: #fee2e2;
      opacity: 0.55;
      transition: opacity 0.2s ease;
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    .forbidden-text {
      font-size: 10px;
      font-weight: 700;
      fill: #b91c1c;
      text-anchor: middle;
      user-select: none;
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    .graph-anchor circle {
      fill: #ffffff;
      stroke: #0284c7;
      stroke-width: 2.5;
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    .anchor-tag {
      font-size: 10px;
      font-weight: 700;
      fill: #0f172a;
      paint-order: stroke;
      stroke: #ffffff;
      stroke-width: 2.5px;
      stroke-linecap: round;
      stroke-linejoin: round;
      user-select: none;
    }
  </style>
</head>
<body>

  
  <div class="header-bar">
    <div class="formula-display" id="formulaMath">
      <span>f(x) = x</span><sup id="expToken"><span class="token-p">2</span>/<span class="token-q">1</span></sup>
      <span>=</span>
      <span id="radicalDisplay">(<sup class="token-q">1</sup>√x)<sup class="token-p">2</sup></span>
    </div>
    <div class="token-legend">
      <span class="legend-tag legend-p">Numerator <b>p</b>: Power</span>
      <span class="legend-tag legend-q">Denominator <b>q</b>: Root</span>
    </div>
  </div>

  
  <div class="tabs-bar" role="tablist">
    <button class="tab-btn active" onclick="setPreset(0)">x²</button>
    <button class="tab-btn" onclick="setPreset(1)">x^(1/2)</button>
    <button class="tab-btn" onclick="setPreset(2)">x^(2/3)</button>
    <button class="tab-btn" onclick="setPreset(3)">x^(-1/2)</button>
  </div>

  
  <div class="content-grid">
    
    <div class="info-panel">
      <div class="info-tile">
        <span class="tile-label">Radical Evaluation</span>
        <div class="radical-text" id="expandedRadical">(¹√x)²</div>
      </div>

      <div class="info-tile">
        <span class="tile-label">Domain & Symmetry</span>
        <div id="domainBadge" class="badge badge-teal">All Real Numbers (Even)</div>
      </div>

      <div class="info-tile">
        <span class="tile-label">Anatomy Rule</span>
        <div class="rule-box" id="ruleText">
          q = 1 (Odd): real for all x.<br>p = 2: squares negatives to positive.
        </div>
      </div>
    </div>

    
    <div class="graph-container">
      <svg id="plotSvg" viewBox="0 0 280 220">
        <defs>
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          <text class="forbidden-text" x="80" y="105">x &lt; 0 undefined in ℝ</text>
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        <line class="axis-line" x1="140" y1="205" x2="140" y2="8" />

        
        
        
        
        <line x1="44" y1="197" x2="44" y2="203" stroke="#94a3b8" stroke-width="1.2" />
        <text class="tick-label" x="44" y="213">-2</text>

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        <text class="tick-label" x="188" y="213">1</text>

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        <text class="tick-label" x="130" y="83">2</text>

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        <text class="tick-label" x="130" y="23">3</text>

        
        <path id="curve" class="curve-path" d="" />

        
        <g id="anchorsGroup"></g>
      </svg>
    </div>
  </div>

  <script>
    const presets = [
      {
        p: 2,
        q: 1,
        expHtml: '<span class="token-p">2</span>/<span class="token-q">1</span>',
        radicalDisplay: '(<sup class="token-q">1</sup>√x)<sup class="token-p">2</sup>',
        expandedRadical: '(x)² = x²',
        badgeText: 'All Real Numbers (Quadrants I & II)',
        badgeClass: 'badge-teal',
        rule: 'q = 1 (Odd root): real everywhere.<br>p = 2: squares negatives to positive.',
        allowNegative: true,
        fn: (x) => Math.pow(x, 2),
        anchors: [
          { x: -1, y: 1, label: '(-1, 1)' },
          { x: 0, y: 0, label: '(0, 0)' },
          { x: 1, y: 1, label: '(1, 1)' }
        ]
      },
      {
        p: 1,
        q: 2,
        expHtml: '<span class="token-p">1</span>/<span class="token-q">2</span>',
        radicalDisplay: '(<sup class="token-q">2</sup>√x)<sup class="token-p">1</sup>',
        expandedRadical: '√x',
        badgeText: 'x ≥ 0 only (Quadrant I)',
        badgeClass: 'badge-amber',
        rule: 'q = 2 (Even root): negative input invalid.<br>Curvature slows as x increases.',
        allowNegative: false,
        fn: (x) => (x >= 0 ? Math.sqrt(x) : NaN),
        anchors: [
          { x: 0, y: 0, label: '(0, 0)' },
          { x: 1, y: 1, label: '(1, 1)' },
          { x: 2.25, y: 1.5, label: '(2.25, 1.5)' }
        ]
      },
      {
        p: 2,
        q: 3,
        expHtml: '<span class="token-p">2</span>/<span class="token-q">3</span>',
        radicalDisplay: '(<sup class="token-q">3</sup>√x)<sup class="token-p">2</sup>',
        expandedRadical: '(∛x)²',
        badgeText: 'All Real Numbers (Symmetric Cusp)',
        badgeClass: 'badge-teal',
        rule: 'q = 3 (Odd root): real for negative x.<br>p = 2 (Even power): creates sharp cusp at 0.',
        allowNegative: true,
        fn: (x) => {
          const cbrt = Math.sign(x) * Math.pow(Math.abs(x), 1/3);
          return Math.pow(cbrt, 2);
        },
        anchors: [
          { x: -1, y: 1, label: '(-1, 1)' },
          { x: 0, y: 0, label: '(0, 0) Cusp' },
          { x: 1, y: 1, label: '(1, 1)' }
        ]
      },
      {
        p: -1,
        q: 2,
        expHtml: '<span class="token-p">-1</span>/<span class="token-q">2</span>',
        radicalDisplay: '1 / <sup class="token-q">2</sup>√x',
        expandedRadical: '1 / √x',
        badgeText: 'x > 0 only (Quadrant I Asymptote)',
        badgeClass: 'badge-amber',
        rule: 'Negative exponent creates vertical asymptote at x = 0.<br>Even root q=2 restricts domain.',
        allowNegative: false,
        fn: (x) => (x > 0.05 ? 1 / Math.sqrt(x) : NaN),
        anchors: [
          { x: 0.25, y: 2, label: '(0.25, 2)' },
          { x: 1, y: 1, label: '(1, 1)' }
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      }
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    function toSvgY(y) {
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    }

    function renderPlot(preset) {
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      forbidden.style.display = preset.allowNegative ? 'none' : 'inline';

      // Path generation
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          isDrawing = false;
          continue;
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        const sx = toSvgX(x);
        const sy = toSvgY(y);

        if (!isDrawing) {
          d += `M ${sx.toFixed(1)} ${sy.toFixed(1)} `;
          isDrawing = true;
        } else {
          d += `L ${sx.toFixed(1)} ${sy.toFixed(1)} `;
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      document.getElementById('curve').setAttribute('d', d);

      // Anchors rendering
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      anchorsGroup.innerHTML = '';

      preset.anchors.forEach(a => {
        const sx = toSvgX(a.x);
        const sy = toSvgY(a.y);
        const isNearZero = Math.abs(a.x) < 0.1;
        const isLeft = a.x < 0;

        const anchorG = document.createElementNS('http://www.w3.org/2000/svg', 'g');
        anchorG.setAttribute('class', 'graph-anchor');

        const circle = document.createElementNS('http://www.w3.org/2000/svg', 'circle');
        circle.setAttribute('cx', sx);
        circle.setAttribute('cy', sy);
        circle.setAttribute('r', '4');

        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
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        text.setAttribute('x', isNearZero ? sx + 6 : (isLeft ? sx - 6 : sx + 6));
        text.setAttribute('y', isNearZero ? sy - 7 : sy - 6);
        text.setAttribute('text-anchor', isLeft ? 'end' : 'start');
        text.textContent = a.label;

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        anchorG.appendChild(text);
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    function setPreset(idx) {
      // Update tabs
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        btn.classList.toggle('active', i === idx);
      });

      const p = presets[idx];

      // Update Math Headers
      document.getElementById('expToken').innerHTML = p.expHtml;
      document.getElementById('radicalDisplay').innerHTML = p.radicalDisplay;
      document.getElementById('expandedRadical').innerHTML = p.expandedRadical;

      // Update Badges & Rules
      const badge = document.getElementById('domainBadge');
      badge.className = `badge ${p.badgeClass}`;
      badge.textContent = p.badgeText;

      document.getElementById('ruleText').innerHTML = p.rule;

      // Update Graph
      renderPlot(p);
    }

    // Initial render
    setPreset(0);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      width: 100%;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e293b;
      background: #ffffff;
      padding: 10px;
    }

    .header {
      text-align: center;
      margin-bottom: 8px;
    }
    .header .formula {
      font-size: 18px;
      font-weight: 700;
      color: #0f172a;
      background: #f1f5f9;
      padding: 4px 12px;
      border-radius: 999px;
      display: inline-block;
    }
    .header .formula span.p { color: #2563eb; }
    .header .formula span.q { color: #059669; }

    /* Decision Grid */
    .tree-grid {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 8px;
      max-width: 640px;
      margin: 0 auto;
    }

    .node-card {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      text-align: center;
      cursor: pointer;
      transition: box-shadow 0.2s ease, background-color 0.2s ease;
    }
    .node-card:hover {
      background: #ffffff;
      box-shadow: 0 4px 12px rgba(0,0,0,0.06);
    }
    .node-card.active {
      background: #ffffff;
      box-shadow: 0 0 0 2px #2563eb, 0 4px 12px rgba(37,99,235,0.12);
    }

    .pill {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      padding: 3px 8px;
      border-radius: 999px;
      margin-bottom: 6px;
      display: inline-block;
    }
    .pill-green { background: #d1fae5; color: #065f46; }
    .pill-blue { background: #dbeafe; color: #1e40af; }
    .pill-purple { background: #ede9fe; color: #5b21b6; }

    .node-title {
      font-size: 13px;
      font-weight: 700;
      color: #0f172a;
      margin-bottom: 2px;
    }

    .badge-sub {
      font-size: 11px;
      color: #64748b;
      margin-bottom: 6px;
      font-weight: 600;
    }

    .plot-box {
      width: 100%;
      height: 90px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 6px;
      position: relative;
      margin-bottom: 6px;
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
    }

    .meta-tag {
      font-size: 11px;
      font-weight: 600;
      color: #334155;
      background: #f1f5f9;
      padding: 2px 6px;
      border-radius: 4px;
      margin-top: 2px;
      width: 100%;
    }

    .flow-nav {
      display: flex;
      justify-content: center;
      gap: 6px;
      margin-top: 8px;
    }
    .flow-btn {
      background: #f8fafc;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      padding: 4px 10px;
      font-size: 12px;
      font-weight: 600;
      color: #334155;
      cursor: pointer;
      transition: background 0.2s ease;
    }
    .flow-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
    }

    @media (max-width: 480px) {
      .tree-grid {
        grid-template-columns: 1fr;
        gap: 6px;
      }
      .plot-box {
        height: 75px;
      }
      .node-card {
        padding: 6px;
      }
    }
  </style>
</head>
<body>

  <div class="header">
    <div class="formula">
      f(x) = x<sup><span class="p">p</span>/<span class="q">q</span></sup>
    </div>
  </div>

  <div class="tree-grid">
    
    <div class="node-card active" id="card-0" onclick="selectBranch(0)">
      <span class="pill pill-green">q is Even</span>
      <div class="node-title">x<sup>1/2</sup> (√x)</div>
      <div class="badge-sub">No negatives under even root</div>
      <div class="plot-box">
        <svg viewBox="-40 -25 80 50" id="svg-0"></svg>
      </div>
      <div class="meta-tag">Domain: [0, ∞)</div>
      <div class="meta-tag">No symmetry</div>
    </div>

    
    <div class="node-card" id="card-1" onclick="selectBranch(1)">
      <span class="pill pill-blue">q Odd • p Even</span>
      <div class="node-title">x<sup>2/3</sup></div>
      <div class="badge-sub">Even symmetry: f(-x) = f(x)</div>
      <div class="plot-box">
        <svg viewBox="-40 -25 80 50" id="svg-1"></svg>
      </div>
      <div class="meta-tag">Domain: ℝ</div>
      <div class="meta-tag">Cusp at Origin</div>
    </div>

    
    <div class="node-card" id="card-2" onclick="selectBranch(2)">
      <span class="pill pill-purple">q Odd • p Odd</span>
      <div class="node-title">x<sup>1/3</sup></div>
      <div class="badge-sub">Odd symmetry: f(-x) = -f(x)</div>
      <div class="plot-box">
        <svg viewBox="-40 -25 80 50" id="svg-2"></svg>
      </div>
      <div class="meta-tag">Domain: ℝ</div>
      <div class="meta-tag">Inflection at Origin</div>
    </div>
  </div>

  <div class="flow-nav">
    <button class="flow-btn active" id="btn-0" onclick="selectBranch(0)">q Even (√x)</button>
    <button class="flow-btn" id="btn-1" onclick="selectBranch(1)">q Odd, p Even (x²⁄³)</button>
    <button class="flow-btn" id="btn-2" onclick="selectBranch(2)">q Odd, p Odd (x¹⁄³)</button>
  </div>

  <script>
    function renderAxes(svg) {
      const grid = document.createElementNS("http://www.w3.org/2000/svg", "g");
      grid.setAttribute("stroke", "#e2e8f0");
      grid.setAttribute("stroke-width", "0.75");
      grid.innerHTML = `
        <line x1="-40" y1="0" x2="40" y2="0" stroke="#94a3b8" stroke-width="1"/>
        <line x1="0" y1="-25" x2="0" y2="25" stroke="#94a3b8" stroke-width="1"/>
      `;
      svg.appendChild(grid);
    }

    function generatePath(fn, xMin, xMax, step = 0.5) {
      let d = "";
      for (let x = xMin; x <= xMax; x += step) {
        const y = fn(x);
        if (isNaN(y) || !isFinite(y)) continue;
        const svgX = x * 10;
        const svgY = -y * 10;
        d += (d === "" ? "M" : "L") + ` ${svgX.toFixed(2)} ${svgY.toFixed(2)}`;
      }
      return d;
    }

    function initPlots() {
      // 1. Root (x^0.5)
      const svg0 = document.getElementById("svg-0");
      renderAxes(svg0);
      const path0 = document.createElementNS("http://www.w3.org/2000/svg", "path");
      path0.setAttribute("d", generatePath(x => Math.sqrt(x), 0, 3.8));
      path0.setAttribute("fill", "none");
      path0.setAttribute("stroke", "#059669");
      path0.setAttribute("stroke-width", "2");
      svg0.appendChild(path0);

      const dot0 = document.createElementNS("http://www.w3.org/2000/svg", "circle");
      dot0.setAttribute("cx", "0");
      dot0.setAttribute("cy", "0");
      dot0.setAttribute("r", "2.5");
      dot0.setAttribute("fill", "#059669");
      svg0.appendChild(dot0);

      // 2. Cusp (x^(2/3))
      const svg1 = document.getElementById("svg-1");
      renderAxes(svg1);
      const path1 = document.createElementNS("http://www.w3.org/2000/svg", "path");
      path1.setAttribute("d", generatePath(x => Math.pow(Math.abs(x), 2/3), -3.6, 3.6, 0.2));
      path1.setAttribute("fill", "none");
      path1.setAttribute("stroke", "#2563eb");
      path1.setAttribute("stroke-width", "2");
      svg1.appendChild(path1);

      const dot1 = document.createElementNS("http://www.w3.org/2000/svg", "circle");
      dot1.setAttribute("cx", "0");
      dot1.setAttribute("cy", "0");
      dot1.setAttribute("r", "2.5");
      dot1.setAttribute("fill", "#2563eb");
      svg1.appendChild(dot1);

      // 3. Cube root (x^(1/3))
      const svg2 = document.getElementById("svg-2");
      renderAxes(svg2);
      const path2 = document.createElementNS("http://www.w3.org/2000/svg", "path");
      path2.setAttribute("d", generatePath(x => Math.sign(x) * Math.pow(Math.abs(x), 1/3), -3.8, 3.8, 0.2));
      path2.setAttribute("fill", "none");
      path2.setAttribute("stroke", "#7c3aed");
      path2.setAttribute("stroke-width", "2");
      svg2.appendChild(path2);

      const dot2 = document.createElementNS("http://www.w3.org/2000/svg", "circle");
      dot2.setAttribute("cx", "0");
      dot2.setAttribute("cy", "0");
      dot2.setAttribute("r", "2.5");
      dot2.setAttribute("fill", "#7c3aed");
      svg2.appendChild(dot2);
    }

    function selectBranch(idx) {
      for (let i = 0; i < 3; i++) {
        document.getElementById(`card-${i}`).classList.toggle("active", i === idx);
        document.getElementById(`btn-${i}`).classList.toggle("active", i === idx);
      }
    }

    initPlots();
  </script>
</body>
</html>

OGM.13.FN.4

Aligned learning resources

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