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  <meta charset="UTF-8">
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  <title>Proof: Linear Growth</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
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      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
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    .btn-step.active {
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    .btn-play {
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      color: #1e2945;
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      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 13px;
      cursor: pointer;
      margin-left: 4px;
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    /* Main Stage */
    .stage {
      position: relative;
      width: 100%;
      max-width: 520px;
      min-height: 220px;
      display: flex;
      align-items: center;
      justify-content: center;
      margin: 12px 0;
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    .step-card {
      position: absolute;
      width: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
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      transition: opacity 0.35s ease, transform 0.35s ease;
      transform: translateY(8px);
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    .step-card.active {
      opacity: 1;
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    .tag {
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      font-weight: 700;
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      margin-bottom: 12px;
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    .math-box {
      background: #f8fafc;
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      .math-text {
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      .btn-step {
        padding: 5px 9px;
        font-size: 12px;
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    .var { font-style: italic; }
    .highlight-m { color: #2563eb; font-weight: 600; }
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    /* Final Success Pill */
    .success-pill {
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      background: #ecfdf5;
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      color: #065f46;
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      font-weight: 600;
      padding: 8px 16px;
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      gap: 6px;
      animation: pulse 2s infinite ease-in-out;
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    @keyframes pulse {
      0%, 100% { transform: scale(1); }
      50% { transform: scale(1.02); }
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      font-weight: 600;
      cursor: pointer;
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  </style>
</head>
<body>

  
  <div class="controls">
    <button class="btn-step active" onclick="goToStep(0)">1</button>
    <button class="btn-step" onclick="goToStep(1)">2</button>
    <button class="btn-step" onclick="goToStep(2)">3</button>
    <button class="btn-step" onclick="goToStep(3)">4</button>
    <button class="btn-step" onclick="goToStep(4)">5</button>
    <button id="playToggle" class="btn-play" onclick="togglePlay()" aria-label="Play/Pause">⏸</button>
  </div>

  
  <div class="stage">

    
    <div class="step-card active" id="step-0">
      <div class="tag">Step 1 · Output difference</div>
      <div class="math-box">
        <div class="math-text">
          Δ<span class="var">y</span> = <span class="var">f</span>(<span class="var">x</span> + <span class="highlight-h var">h</span>) − <span class="var">f</span>(<span class="var">x</span>)
        </div>
      </div>
    </div>

    
    <div class="step-card" id="step-1">
      <div class="tag">Step 2 · Apply linear form</div>
      <div class="math-box">
        <div class="math-text">
          Δ<span class="var">y</span> = [<span class="highlight-m var">m</span>(<span class="var">x</span> + <span class="highlight-h var">h</span>) + <span class="highlight-b var">b</span>] − [<span class="highlight-m var">m</span><span class="var">x</span> + <span class="highlight-b var">b</span>]
        </div>
      </div>
    </div>

    
    <div class="step-card" id="step-2">
      <div class="tag">Step 3 · Expand terms</div>
      <div class="math-box">
        <div class="math-text">
          Δ<span class="var">y</span> = <span class="var">m</span><span class="var">x</span> + <span class="highlight-m var">m</span><span class="highlight-h var">h</span> + <span class="var">b</span> − <span class="var">m</span><span class="var">x</span> − <span class="var">b</span>
        </div>
      </div>
    </div>

    
    <div class="step-card" id="step-3">
      <div class="tag">Step 4 · Cancel opposites</div>
      <div class="math-box">
        <div class="math-text">
          Δ<span class="var">y</span> = <span class="strike"><span class="var">m</span><span class="var">x</span></span> + <span class="highlight-m var">m</span><span class="highlight-h var">h</span> + <span class="strike"><span class="var">b</span></span> − <span class="strike"><span class="var">m</span><span class="var">x</span></span> − <span class="strike"><span class="var">b</span></span>
        </div>
      </div>
    </div>

    
    <div class="step-card" id="step-4">
      <div class="tag">Step 5 · Final result</div>
      <div class="math-box" style="border-color: #86efac; background: #f0fdf4;">
        <div class="math-text" style="font-size: 26px;">
          Δ<span class="var">y</span> = <span class="highlight-m var">m</span><span class="highlight-h var">h</span>
        </div>
      </div>
      <div class="success-pill">
        <span>✓</span> Notice <span class="var">x</span> completely vanishes!
      </div>
    </div>

  </div>

  
  <div class="nav-bar">
    <button class="nav-btn" onclick="prevStep()">← Prev</button>
    <button class="nav-btn" onclick="nextStep()">Next →</button>
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    startTimer();
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</body>
</html>

Proving Linear Functions Grow by Equal Differences Over Equal Intervals

<!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: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: #ffffff;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      justify-content: center;
      min-height: 100%;
    }
    .header {
      display: flex;
      justify-content: space-between;
      align-items: center;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 14px;
      font-size: 15px;
      font-weight: 600;
    }
    .formula-tag {
      color: #0284c7;
    }
    .step-tag {
      background: #e0f2fe;
      color: #0369a1;
      padding: 2px 8px;
      border-radius: 6px;
      font-size: 13px;
    }
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      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
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    @media (max-width: 440px) {
      .cards-grid {
        grid-template-columns: 1fr;
      }
    }
    .calc-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 10px 12px;
      display: flex;
      flex-direction: column;
      gap: 8px;
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    .card-title {
      font-size: 14px;
      font-weight: 700;
      color: #475569;
      display: flex;
      align-items: center;
      justify-content: space-between;
      border-bottom: 1px dashed #e2e8f0;
      padding-bottom: 6px;
    }
    .badge {
      font-size: 12px;
      font-weight: 600;
      padding: 2px 6px;
      border-radius: 4px;
      background: #f1f5f9;
      color: #334155;
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      background: #f8fafc;
      padding: 5px 8px;
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    .diff-box {
      margin-top: 2px;
      background: #eff6ff;
      border: 1px solid #bfdbfe;
      border-radius: 6px;
      padding: 8px 10px;
      text-align: center;
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    .diff-label {
      font-size: 12px;
      font-weight: 600;
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      justify-content: center;
      gap: 6px;
      font-size: 13.5px;
      font-weight: 600;
      color: #166534;
      text-align: center;
    }
    .banner-math {
      font-family: ui-monospace, SFMono-Regular, Menlo, monospace;
      font-weight: 700;
      color: #15803d;
    }
  </style>
</head>
<body>

  <div class="header">
    <span class="formula-tag">f(x) = −4x + 9</span>
    <span class="step-tag">Step h = +0.5</span>
  </div>

  <div class="cards-grid">
    
    <div class="calc-card">
      <div class="card-title">
        <span>Start at x = 1</span>
        <span class="badge">Interval 1</span>
      </div>
      <div class="eval-row">
        <span>f(1.0)</span>
        <span>= 5</span>
      </div>
      <div class="eval-row">
        <span>f(1.5)</span>
        <span>= 3</span>
      </div>
      <div class="diff-box">
        <div class="diff-label">Change in y (Δy)</div>
        <div class="diff-math">3 − 5 = <span class="result-highlight">−2</span></div>
      </div>
    </div>

    
    <div class="calc-card">
      <div class="card-title">
        <span>Start at x = 10</span>
        <span class="badge">Interval 2</span>
      </div>
      <div class="eval-row">
        <span>f(10.0)</span>
        <span>= −31</span>
      </div>
      <div class="eval-row">
        <span>f(10.5)</span>
        <span>= −33</span>
      </div>
      <div class="diff-box">
        <div class="diff-label">Change in y (Δy)</div>
        <div class="diff-math">−33 − (−31) = <span class="result-highlight">−2</span></div>
      </div>
    </div>
  </div>

  <div class="banner">
    <span>Formula:</span>
    <span class="banner-math">Δy = (−4)(0.5) = −2</span>
    <span>in both cases</span>
  </div>

</body>
</html>

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<html lang="en">
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      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      padding: 10px;
      user-select: none;
    }
    .controls {
      display: flex;
      gap: 8px;
      margin-bottom: 8px;
      flex-wrap: wrap;
      justify-content: center;
    }
    .btn {
      background: #f8fafc;
      border: 1.5px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 12px;
      font-size: 14px;
      font-weight: 600;
      color: #1e2945;
      cursor: pointer;
      transition: background 0.2s, border-color 0.2s, box-shadow 0.2s;
    }
    .btn.active {
      background: #eff6ff;
      border-color: #3b82f6;
      color: #1d4ed8;
      box-shadow: 0 0 0 1px #3b82f6;
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      width: 100%;
      max-width: 480px;
      position: relative;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
    }
    .grid-line {
      stroke: #f1f5f9;
      stroke-width: 1;
    }
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      stroke: #94a3b8;
      stroke-width: 1.5;
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    .axis-label {
      font-size: 13px;
      fill: #64748b;
      font-weight: 500;
      text-anchor: middle;
      dominant-baseline: middle;
    }
    .func-line {
      stroke: #3b82f6;
      stroke-width: 3.5;
      stroke-linecap: round;
    }
    .step-group {
      transition: opacity 0.3s ease;
    }
    .step-group.dimmed {
      opacity: 0.2;
    }
    .step-dx {
      stroke: #059669;
      stroke-width: 2.5;
    }
    .step-dy {
      stroke: #10b981;
      stroke-width: 2.5;
      stroke-dasharray: 4 3;
    }
    .step-bracket {
      stroke: #059669;
      stroke-width: 1.5;
      fill: none;
    }
    .tag-text {
      font-size: 13px;
      font-weight: 700;
      fill: #065f46;
      text-anchor: middle;
      dominant-baseline: central;
    }
    .pt {
      fill: #1e2945;
      stroke: #ffffff;
      stroke-width: 2;
    }
    .banner {
      margin-top: 6px;
      font-size: 13px;
      font-weight: 600;
      color: #1e2945;
      text-align: center;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 12px;
      max-width: 480px;
      width: 100%;
    }
    .highlight {
      color: #059669;
      font-weight: 700;
    }
  </style>
</head>
<body>

  <div class="controls">
    <button class="btn" onclick="showStep(1)" id="b1">Step 1</button>
    <button class="btn" onclick="showStep(2)" id="b2">Step 2</button>
    <button class="btn" onclick="showStep(3)" id="b3">Step 3</button>
    <button class="btn active" onclick="showStep(0)" id="b0">Compare All</button>
  </div>

  <div class="graph-container">
    <svg viewBox="0 0 460 400" id="plot">
      
      <g id="grid"></g>

      
      <line x1="45" y1="350" x2="445" y2="350" class="axis" />
      <line x1="45" y1="350" x2="45" y2="20" class="axis" />

      
      <g id="ticks"></g>

      
      
      <line x1="45" y1="330" x2="425" y2="50" class="func-line" />

      
      
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        <line x1="45" y1="330" x2="155" y2="330" class="step-dx" />
        <line x1="155" y1="330" x2="155" y2="250" class="step-dy" />
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        <circle cx="155" cy="250" r="4.5" class="pt" />
        
        <rect x="75" y="334" width="50" height="18" rx="4" fill="#d1fae5" />
        <text x="100" y="343" class="tag-text">Δx = 2</text>
        <rect x="162" y="281" width="50" height="18" rx="4" fill="#d1fae5" />
        <text x="187" y="290" class="tag-text">Δy = 4</text>
      </g>

      
      <g id="step2" class="step-group">
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        <rect x="185" y="254" width="50" height="18" rx="4" fill="#d1fae5" />
        <text x="210" y="263" class="tag-text">Δx = 2</text>
        <rect x="272" y="201" width="50" height="18" rx="4" fill="#d1fae5" />
        <text x="297" y="210" class="tag-text">Δy = 4</text>
      </g>

      
      <g id="step3" class="step-group">
        <line x1="265" y1="170" x2="375" y2="170" class="step-dx" />
        <line x1="375" y1="170" x2="375" y2="90" class="step-dy" />
        <circle cx="375" cy="90" r="4.5" class="pt" />
        
        <rect x="295" y="174" width="50" height="18" rx="4" fill="#d1fae5" />
        <text x="320" y="183" class="tag-text">Δx = 2</text>
        <rect x="382" y="121" width="50" height="18" rx="4" fill="#d1fae5" />
        <text x="407" y="130" class="tag-text">Δy = 4</text>
      </g>

      
      <rect x="48" y="24" width="108" height="24" rx="6" fill="#eff6ff" stroke="#bfdbfe" stroke-width="1" />
      <text x="102" y="36" fill="#1d4ed8" font-size="13" font-weight="700" text-anchor="middle" dominant-baseline="central">f(x) = 2x + 1</text>
    </svg>
  </div>

  <div class="banner" id="banner">
    Equal steps (<span class="highlight">Δx = 2</span>) always produce identical vertical changes (<span class="highlight">Δy = 4</span>).
  </div>

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    const ticksG = document.getElementById('ticks');

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    // y: 0 to 16 -> px: 350 - y * 20
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    function showStep(idx) {
      document.querySelectorAll('.btn').forEach(b => b.classList.remove('active'));
      document.getElementById('b' + idx).classList.add('active');

      const s1 = document.getElementById('step1');
      const s2 = document.getElementById('step2');
      const s3 = document.getElementById('step3');
      const banner = document.getElementById('banner');

      if (idx === 0) {
        s1.classList.remove('dimmed');
        s2.classList.remove('dimmed');
        s3.classList.remove('dimmed');
        banner.innerHTML = 'Equal steps (<span class="highlight">Δx = 2</span>) always produce identical vertical changes (<span class="highlight">Δy = 4</span>).';
      } else {
        s1.classList.toggle('dimmed', idx !== 1);
        s2.classList.toggle('dimmed', idx !== 2);
        s3.classList.toggle('dimmed', idx !== 3);
        const ranges = [
          '',
          'x = 0 to 2: Δx = 2 → Δy = 2(2) = 4',
          'x = 2 to 4: Δx = 2 → Δy = 2(2) = 4',
          'x = 4 to 6: Δx = 2 → Δy = 2(2) = 4'
        ];
        banner.innerHTML = `<span class="highlight">${ranges[idx]}</span> (m = 2)`;
      }
    }
  </script>
</body>
</html>

OGM.12.EXP.1

Aligned learning resources

Related diagrams