Advanced

Tutor model

i62 💰:9.4

i52 💰:7.9

🏆 i67 💰:62.5

i70 💰:125.0

i68 💰:125.0

🏆 i66 💰:75.0

i65 💰:125.0

i63 💰:62.5

i61 💰:75.0

i60 💰:9.4

i60 💰:37.5

i58 💰:62.5

i57 💰:22.5

i57 💰:37.5

i56 💰:45.0

⚡ i55 💰:11.25

i55 💰:37.5

i54 💰:62.5

i52 💰:35.0

i51 💰:37.5

i49 💰:15.0

i49 💰:62.5

i48 💰:11.3

i47 💰:7.5

i44 💰:25.0

i43 💰:37.5

i42 💰:187.5

i40 💰:5.0

i39 💰:100.0

i38 💰:37.5

i37 💰:11.0

i36 💰:187.5

i34 💰:12.5

i33 💰:37.5

i30 💰:1.0

i29 💰:1.5

i28 💰:150.0

i23 💰:20.0

i20 💰:4.0

i18 💰:1.0

i16 💰:25.0

i13 💰:1.5

Keep session private

What would you like to learn?

Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

Lesson generation model

System default

i58 💰:18.8

i31 💰:11.0

i25 💰:31.3

Tutoring abilities

When you toss a basketball toward a hoop, its path isn't a straight line — it arcs up, reaches a peak, and curves smoothly back down.

That signature U-shaped curve is modeled by a **quadratic function**, one of the most useful tools in algebra.

What makes an equation quadratic in the first place?

A **quadratic function** is a polynomial function where the highest exponent of the variable is $2$ (known as degree $2$).

Its **standard form** is written as $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and the leading coefficient $a \neq 0$.

<!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">
  <title>Quadratic Function Explorer</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      user-select: none;
    }
    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 12px;
      min-height: 100%;
    }
    .main-layout {
      width: 100%;
      max-width: 360px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
    }
    .equation-badge {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 20px;
      padding: 6px 16px;
      font-size: 17px;
      font-weight: 600;
      color: #1e2945;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .coef-a {
      color: #22b7ff;
      font-weight: 700;
    }
    .state-tag {
      font-size: 13px;
      font-weight: 600;
      padding: 2px 8px;
      border-radius: 6px;
      margin-left: 6px;
      background: #e0f2fe;
      color: #0369a1;
      transition: background 0.2s, color 0.2s;
    }
    .state-tag.linear {
      background: #fef3c7;
      color: #92400e;
    }
    .graph-panel {
      width: 100%;
      aspect-ratio: 1 / 0.85;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      position: relative;
      overflow: hidden;
      box-shadow: 0 2px 6px rgba(0,0,0,0.04);
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
    }
    .controls {
      width: 100%;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }
    .slider-row {
      display: flex;
      align-items: center;
      gap: 10px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 6px 12px;
    }
    .slider-label {
      font-size: 14px;
      font-weight: 700;
      color: #22b7ff;
      min-width: 48px;
    }
    input[type="range"] {
      flex: 1;
      accent-color: #22b7ff;
      cursor: pointer;
      height: 6px;
    }
    .btn-row {
      display: flex;
      gap: 6px;
      width: 100%;
    }
    .preset-btn {
      flex: 1;
      padding: 7px 4px;
      font-size: 12px;
      font-weight: 600;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      color: #1e2945;
      cursor: pointer;
      text-align: center;
      transition: all 0.15s ease;
    }
    .preset-btn:active {
      transform: translateY(1px);
    }
    .preset-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
    }
  </style>
</head>
<body>
  <div class="main-layout">
    <div class="equation-badge">
      <span>f(x) = </span>
      <span class="coef-a" id="eqA">1</span>
      <span>x²</span>
      <span class="state-tag" id="curveTag">Opens Up</span>
    </div>

    <div class="graph-panel">
      <svg id="graphSvg" viewBox="-160 -130 320 260">
        
        <g stroke="#f1f5f9" stroke-width="1">
          <line x1="-160" y1="-100" x2="160" y2="-100" />
          <line x1="-160" y1="-75" x2="160" y2="-75" />
          <line x1="-160" y1="-50" x2="160" y2="-50" />
          <line x1="-160" y1="-25" x2="160" y2="-25" />
          <line x1="-160" y1="25" x2="160" y2="25" />
          <line x1="-160" y1="50" x2="160" y2="50" />
          <line x1="-160" y1="75" x2="160" y2="75" />
          <line x1="-160" y1="100" x2="160" y2="100" />

          <line x1="-125" y1="-130" x2="-125" y2="130" />
          <line x1="-100" y1="-130" x2="-100" y2="130" />
          <line x1="-75" y1="-130" x2="-75" y2="130" />
          <line x1="-50" y1="-130" x2="-50" y2="130" />
          <line x1="-25" y1="-130" x2="-25" y2="130" />
          <line x1="25" y1="-130" x2="25" y2="130" />
          <line x1="50" y1="-130" x2="50" y2="130" />
          <line x1="75" y1="-130" x2="75" y2="130" />
          <line x1="100" y1="-130" x2="100" y2="130" />
          <line x1="125" y1="-130" x2="125" y2="130" />
        </g>

        
        <g stroke="#cbd5e1" stroke-width="1.5">
          <line x1="-160" y1="0" x2="160" y2="0" />
          <line x1="0" y1="-130" x2="0" y2="130" />
        </g>

        
        <path id="curvePath" fill="none" stroke="#22b7ff" stroke-width="3" stroke-linecap="round" stroke-linejoin="round" />

        
        <circle id="vertexDot" cx="0" cy="0" r="4.5" fill="#1e2945" />
      </svg>
    </div>

    <div class="controls">
      <div class="slider-row">
        <span class="slider-label" id="sliderVal">a = 1.0</span>
        <input type="range" id="aSlider" min="-2" max="2" step="0.1" value="1">
      </div>

      <div class="btn-row">
        <button class="preset-btn active" id="btnPos" onclick="setA(1)">a &gt; 0 ∪</button>
        <button class="preset-btn" id="btnZero" onclick="setA(0)">a = 0 (Line)</button>
        <button class="preset-btn" id="btnNeg" onclick="setA(-1)">a &lt; 0 ∩</button>
      </div>
    </div>
  </div>

  <script>
    const slider = document.getElementById('aSlider');
    const curvePath = document.getElementById('curvePath');
    const eqA = document.getElementById('eqA');
    const sliderVal = document.getElementById('sliderVal');
    const curveTag = document.getElementById('curveTag');
    const btnPos = document.getElementById('btnPos');
    const btnZero = document.getElementById('btnZero');
    const btnNeg = document.getElementById('btnNeg');

    // Scale: 25px per unit
    const SCALE = 25;

    function render(a) {
      // Generate SVG path points
      const points = [];
      const step = 0.1;
      for (let x = -6.5; x <= 6.5; x += step) {
        const y = a * x * x;
        const svgX = x * SCALE;
        const svgY = -y * SCALE; // inverted SVG Y
        if (svgY >= -140 && svgY <= 140) {
          points.push(`${svgX.toFixed(1)},${svgY.toFixed(1)}`);
        }
      }

      if (points.length > 0) {
        curvePath.setAttribute('d', 'M ' + points.join(' L '));
      }

      // Update Text & Labels
      sliderVal.textContent = `a = ${a.toFixed(1)}`;
      
      if (a === 1) {
        eqA.textContent = '';
      } else if (a === -1) {
        eqA.textContent = '-';
      } else {
        eqA.textContent = a.toFixed(1);
      }

      if (Math.abs(a) < 0.01) {
        eqA.textContent = '0·';
        curveTag.textContent = 'Linear (a = 0)';
        curveTag.className = 'state-tag linear';
        curvePath.setAttribute('stroke', '#f59e0b');
      } else if (a > 0) {
        curveTag.textContent = 'Opens Up';
        curveTag.className = 'state-tag';
        curvePath.setAttribute('stroke', '#22b7ff');
      } else {
        curveTag.textContent = 'Opens Down';
        curveTag.className = 'state-tag';
        curvePath.setAttribute('stroke', '#6366f1');
      }

      // Sync button states
      btnPos.classList.toggle('active', a > 0.05);
      btnZero.classList.toggle('active', Math.abs(a) < 0.05);
      btnNeg.classList.toggle('active', a < -0.05);
    }

    function setA(val) {
      slider.value = val;
      render(parseFloat(val));
    }

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

    // Initial render
    render(1);
  </script>
</body>
</html>

If $a = 0$, the $x^2$ term disappears and you get a linear function, which produces a straight line instead of a curve.

What does this curve actually look like on a coordinate plane?

The graph of any quadratic function is a smooth, symmetrical curve called a **parabola**.

Every parabola has a turning point called the **vertex**, which is either the highest point (maximum) or lowest point (minimum) on the graph.

<!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">
  <title>Quadratic Vertex: Minimum and Maximum</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      color: #1e2945;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 8px;
      user-select: none;
    }
    .controls {
      display: flex;
      gap: 8px;
      margin-bottom: 10px;
      z-index: 10;
    }
    .toggle-btn {
      padding: 6px 14px;
      font-size: 14px;
      font-weight: 600;
      border: 1px solid #d1d5db;
      border-radius: 20px;
      background: #ffffff;
      color: #4b5563;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
      transition: background 0.2s ease, color 0.2s ease, border-color 0.2s ease;
    }
    .toggle-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
    }
    .graph-container {
      position: relative;
      width: 100%;
      max-width: 440px;
      aspect-ratio: 1 / 1;
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 4px 12px rgba(0,0,0,0.04);
      border: 1px solid #e5e7eb;
      overflow: hidden;
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
    }
    .grid-line {
      stroke: #f1f5f9;
      stroke-width: 1;
    }
    .axis {
      stroke: #94a3b8;
      stroke-width: 1.5;
    }
    .axis-arrow {
      fill: #94a3b8;
    }
    .parabola-path {
      fill: none;
      stroke: #22b7ff;
      stroke-width: 3.5;
      stroke-linecap: round;
      stroke-linejoin: round;
      transition: d 0.4s cubic-bezier(0.4, 0, 0.2, 1);
    }
    .sym-line {
      stroke: #6366f1;
      stroke-width: 1.5;
      stroke-dasharray: 4 4;
      transition: x1 0.4s ease, x2 0.4s ease;
    }
    .vertex-pulse {
      fill: #22b7ff;
      opacity: 0.25;
      transition: cx 0.4s cubic-bezier(0.4, 0, 0.2, 1), cy 0.4s cubic-bezier(0.4, 0, 0.2, 1);
    }
    .vertex-dot {
      fill: #0284c7;
      stroke: #ffffff;
      stroke-width: 2.5;
      transition: cx 0.4s cubic-bezier(0.4, 0, 0.2, 1), cy 0.4s cubic-bezier(0.4, 0, 0.2, 1);
    }
    .tag {
      font-size: 13px;
      font-weight: 700;
      letter-spacing: -0.2px;
      transition: transform 0.4s cubic-bezier(0.4, 0, 0.2, 1), opacity 0.3s ease;
    }
    .tag-bg {
      fill: #ffffff;
      stroke: #e2e8f0;
      stroke-width: 1;
      rx: 6;
      filter: drop-shadow(0 2px 4px rgba(0,0,0,0.06));
    }
    .tag-text-main {
      fill: #0f172a;
    }
    .tag-text-sub {
      fill: #0284c7;
      font-weight: 600;
      font-size: 11px;
    }
  </style>
</head>
<body>

  <div class="controls">
    <button class="toggle-btn active" id="btnMin">
      <span>∪</span> Minimum (Opens Up)
    </button>
    <button class="toggle-btn" id="btnMax">
      <span>∩</span> Maximum (Opens Down)
    </button>
  </div>

  <div class="graph-container">
    <svg viewBox="-5 -5 10 10" preserveAspectRatio="xMidYMid meet">
      
      <g id="grid"></g>

      
      <line x1="-5" y1="0" x2="5" y2="0" class="axis" />
      <line x1="0" y1="-5" x2="0" y2="5" class="axis" />

      
      <line id="symLine" x1="0" y1="-5" x2="0" y2="5" class="sym-line" />

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

      
      <circle id="vertexPulse" class="vertex-pulse" cx="0" cy="0" r="0.6" />
      <circle id="vertexDot" class="vertex-dot" cx="0" cy="0" r="0.25" />

      
      <g id="labelGroup" class="tag" transform="translate(0, 0)">
        <rect class="tag-bg" x="-38" y="-36" width="76" height="30" />
        <text class="tag-text-main" x="0" y="-21" text-anchor="middle" font-size="9.5">Vertex</text>
        <text id="vertexType" class="tag-text-sub" x="0" y="-10" text-anchor="middle" font-size="8">Lowest Point</text>
      </g>
    </svg>
  </div>

  <script>
    // Build background grid (-5 to +5)
    const grid = document.getElementById('grid');
    for (let i = -5; i <= 5; i++) {
      if (i !== 0) {
        // Vertical
        const vLine = document.createElementNS('http://www.w3.org/2000/svg', 'line');
        vLine.setAttribute('x1', i);
        vLine.setAttribute('y1', -5);
        vLine.setAttribute('x2', i);
        vLine.setAttribute('y2', 5);
        vLine.setAttribute('class', 'grid-line');
        grid.appendChild(vLine);
        
        // Horizontal
        const hLine = document.createElementNS('http://www.w3.org/2000/svg', 'line');
        hLine.setAttribute('x1', -5);
        hLine.setAttribute('y1', i);
        hLine.setAttribute('x2', 5);
        hLine.setAttribute('y2', i);
        hLine.setAttribute('class', 'grid-line');
        grid.appendChild(hLine);
      }
    }

    // State & Math calculation
    // SVG coordinates: y increases downward.
    // Mathematical (x, y_math) -> SVG (x, -y_math)
    const parabola = document.getElementById('parabola');
    const vertexPulse = document.getElementById('vertexPulse');
    const vertexDot = document.getElementById('vertexDot');
    const labelGroup = document.getElementById('labelGroup');
    const vertexType = document.getElementById('vertexType');
    const btnMin = document.getElementById('btnMin');
    const btnMax = document.getElementById('btnMax');

    function computeParabolaPath(a, h, k) {
      let d = '';
      const step = 0.1;
      let first = true;
      for (let x = -4.5; x <= 4.5; x += step) {
        const yMath = a * Math.pow(x - h, 2) + k;
        const ySvg = -yMath; // Invert for SVG
        if (first) {
          d += `M ${x.toFixed(2)} ${ySvg.toFixed(2)}`;
          first = false;
        } else {
          d += ` L ${x.toFixed(2)} ${ySvg.toFixed(2)}`;
        }
      }
      return d;
    }

    function setParabolaState(mode) {
      if (mode === 'min') {
        // y = 0.55 * x^2 - 3.2 (Opens UP, vertex at 0, -3.2)
        const a = 0.55, h = 0, k = -3.2;
        parabola.setAttribute('d', computeParabolaPath(a, h, k));
        
        const svgVy = -k; // 3.2
        vertexDot.setAttribute('cx', h);
        vertexDot.setAttribute('cy', svgVy);
        vertexPulse.setAttribute('cx', h);
        vertexPulse.setAttribute('cy', svgVy);

        // Position label above/below vertex
        labelGroup.setAttribute('transform', `translate(${h}, ${svgVy + 0.9}) scale(0.04)`);
        vertexType.textContent = 'Lowest Point';

        btnMin.classList.add('active');
        btnMax.classList.remove('active');
      } else {
        // y = -0.55 * x^2 + 3.2 (Opens DOWN, vertex at 0, 3.2)
        const a = -0.55, h = 0, k = 3.2;
        parabola.setAttribute('d', computeParabolaPath(a, h, k));

        const svgVy = -k; // -3.2
        vertexDot.setAttribute('cx', h);
        vertexDot.setAttribute('cy', svgVy);
        vertexPulse.setAttribute('cx', h);
        vertexPulse.setAttribute('cy', svgVy);

        // Position label above vertex
        labelGroup.setAttribute('transform', `translate(${h}, ${svgVy - 0.25}) scale(0.04)`);
        vertexType.textContent = 'Highest Point';

        btnMax.classList.add('active');
        btnMin.classList.remove('active');
      }
    }

    btnMin.addEventListener('click', () => setParabolaState('min'));
    btnMax.addEventListener('click', () => setParabolaState('max'));

    // Initialize
    setParabolaState('min');
  </script>
</body>
</html>

The vertical fold line that splits the parabola into two mirror images is the **axis of symmetry**, and it always passes straight through the vertex at $x = h$.

How can you figure out where all these key landmarks are just by reading the equation?

## Reading Key Features From the Equation

The sign of $a$ tells you which way the parabola opens: if $a > 0$, it opens upward like a cup (minimum vertex); if $a < 0$, it opens downward like a hill (maximum vertex).

Because parabolas are perfectly symmetric, the vertex sits midway between any two symmetric points, giving us the **axis of symmetry formula** $x = -\frac{b}{2a}$.

<!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">
  <title>Axis of Symmetry & Vertex</title>
  <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/katex/0.16.8/katex.min.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/katex/0.16.8/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100vh;
      padding: 12px 16px;
      user-select: none;
    }
    .top-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 16px;
      width: 100%;
      max-width: 500px;
    }
    .badge {
      background: #ffffff;
      padding: 6px 14px;
      border-radius: 20px;
      box-shadow: 0 2px 6px rgba(0,0,0,0.06);
      font-size: 14px;
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .badge.axis {
      border: 1.5px solid #22b7ff;
      color: #0077b6;
    }
    .badge.vertex {
      border: 1.5px solid #ff4d6d;
      color: #d90429;
    }
    .graph-container {
      width: 100%;
      max-width: 480px;
      aspect-ratio: 4 / 3;
      position: relative;
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 3px 10px rgba(0,0,0,0.05);
      overflow: hidden;
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
    }
    .controls {
      width: 100%;
      max-width: 480px;
      display: flex;
      align-items: center;
      gap: 12px;
      background: #ffffff;
      padding: 8px 16px;
      border-radius: 12px;
      box-shadow: 0 2px 6px rgba(0,0,0,0.05);
    }
    .controls span {
      font-size: 13px;
      font-weight: 600;
      color: #6b7280;
      white-space: nowrap;
    }
    input[type="range"] {
      flex: 1;
      accent-color: #22b7ff;
      cursor: pointer;
      height: 6px;
    }
  </style>
</head>
<body>

  <div class="top-bar">
    <div class="badge axis" id="axisBadge"></div>
    <div class="badge vertex" id="vertexBadge"></div>
  </div>

  <div class="graph-container">
    <svg id="svgGraph" viewBox="0 0 400 300">
      <defs>
        
        <marker id="arr-left" markerWidth="6" markerHeight="6" refX="2" refY="3" orient="auto">
          <path d="M5,0 L1,3 L5,6" fill="none" stroke="#22b7ff" stroke-width="1.5" stroke-linecap="round"/>
        </marker>
        <marker id="arr-right" markerWidth="6" markerHeight="6" refX="4" refY="3" orient="auto">
          <path d="M1,0 L5,3 L1,6" fill="none" stroke="#22b7ff" stroke-width="1.5" stroke-linecap="round"/>
        </marker>
      </defs>

      
      <g stroke="#f0f2f5" stroke-width="1">
        <line x1="40" y1="0" x2="40" y2="300" />
        <line x1="80" y1="0" x2="80" y2="300" />
        <line x1="120" y1="0" x2="120" y2="300" />
        <line x1="160" y1="0" x2="160" y2="300" />
        <line x1="200" y1="0" x2="200" y2="300" />
        <line x1="240" y1="0" x2="240" y2="300" />
        <line x1="280" y1="0" x2="280" y2="300" />
        <line x1="320" y1="0" x2="320" y2="300" />
        <line x1="360" y1="0" x2="360" y2="300" />

        <line x1="0" y1="50" x2="400" y2="50" />
        <line x1="0" y1="100" x2="400" y2="100" />
        <line x1="0" y1="150" x2="400" y2="150" />
        <line x1="0" y1="200" x2="400" y2="200" />
        <line x1="0" y1="250" x2="400" y2="250" />
      </g>

      
      <line x1="0" y1="260" x2="400" y2="260" stroke="#cbd5e1" stroke-width="1.5" />
      <line x1="60" y1="0" x2="60" y2="300" stroke="#cbd5e1" stroke-width="1.5" />

      
      <line id="axisLine" x1="200" y1="15" x2="200" y2="285" stroke="#22b7ff" stroke-width="2" stroke-dasharray="6,4" />

      
      <path id="parabolaPath" fill="none" stroke="#1e2945" stroke-width="3" stroke-linecap="round" />

      
      <line id="secantLine" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="3,3" />

      
      <line id="distLeft" stroke="#22b7ff" stroke-width="2" marker-start="url(#arr-left)" marker-end="url(#arr-right)" />
      <line id="distRight" stroke="#22b7ff" stroke-width="2" marker-start="url(#arr-left)" marker-end="url(#arr-right)" />

      
      <text id="lblLeft" fill="#0077b6" font-size="12" font-weight="700" text-anchor="middle" dominant-baseline="auto">d</text>
      <text id="lblRight" fill="#0077b6" font-size="12" font-weight="700" text-anchor="middle" dominant-baseline="auto">d</text>

      
      <circle id="ptLeft" r="6" fill="#22b7ff" stroke="#ffffff" stroke-width="2" />
      <circle id="ptRight" r="6" fill="#22b7ff" stroke="#ffffff" stroke-width="2" />

      
      <circle id="ptVertex" cx="200" cy="210" r="7" fill="#ff4d6d" stroke="#ffffff" stroke-width="2.5" />

      
      <text id="lblVertex" x="200" y="234" fill="#d90429" font-size="12" font-weight="bold" text-anchor="middle">(h, k)</text>
      <text id="lblAxisTop" x="200" y="14" fill="#0077b6" font-size="11" font-weight="bold" text-anchor="middle">x = h</text>
    </svg>
  </div>

  <div class="controls">
    <span>Height:</span>
    <input type="range" id="heightSlider" min="20" max="130" value="80" step="1">
  </div>

  <script>
    // Parabola math: vertex at (x0, y0) = (200, 210) in SVG pixels.
    // SVG y grows downward, so y_svg = y0 - a * (x_svg - x0)^2
    const x0 = 200;
    const y0 = 210;
    const a = 0.012; // curve scale factor

    // Draw static parabola path
    function buildParabola() {
      let d = "";
      for (let x = 70; x <= 330; x += 2) {
        const y = y0 - a * Math.pow(x - x0, 2);
        d += (x === 70 ? `M ${x} ${y}` : ` L ${x} ${y}`);
      }
      document.getElementById('parabolaPath').setAttribute('d', d);
    }

    function update(dy) {
      // dy is vertical distance above vertex
      const curY = y0 - dy;
      const dx = Math.sqrt(dy / a);

      const xLeft = x0 - dx;
      const xRight = x0 + dx;

      // Symmetric points
      const ptL = document.getElementById('ptLeft');
      const ptR = document.getElementById('ptRight');
      ptL.setAttribute('cx', xLeft);
      ptL.setAttribute('cy', curY);
      ptR.setAttribute('cx', xRight);
      ptR.setAttribute('cy', curY);

      // Horizontal dashed line
      const secLine = document.getElementById('secantLine');
      secLine.setAttribute('x1', Math.max(30, xLeft - 20));
      secLine.setAttribute('y1', curY);
      secLine.setAttribute('x2', Math.min(370, xRight + 20));
      secLine.setAttribute('y2', curY);

      // Distance indicator lines & labels
      const dL = document.getElementById('distLeft');
      const dR = document.getElementById('distRight');
      const lblL = document.getElementById('lblLeft');
      const lblR = document.getElementById('lblRight');

      const gap = 12;
      const arrowY = curY - gap;

      dL.setAttribute('x1', xLeft + 4);
      dL.setAttribute('y1', arrowY);
      dL.setAttribute('x2', x0 - 4);
      dL.setAttribute('y2', arrowY);

      dR.setAttribute('x1', x0 + 4);
      dR.setAttribute('y1', arrowY);
      dR.setAttribute('x2', xRight - 4);
      dR.setAttribute('y2', arrowY);

      lblL.setAttribute('x', (xLeft + x0) / 2);
      lblL.setAttribute('y', arrowY - 4);
      lblR.setAttribute('x', (x0 + xRight) / 2);
      lblR.setAttribute('y', arrowY - 4);
    }

    const slider = document.getElementById('heightSlider');
    slider.addEventListener('input', (e) => {
      update(parseFloat(e.target.value));
    });

    // Render KaTeX badges
    window.addEventListener('DOMContentLoaded', () => {
      buildParabola();
      update(parseFloat(slider.value));

      if (window.katex) {
        katex.render("x = -\\frac{b}{2a}", document.getElementById('axisBadge'), { throwOnError: false });
        katex.render("(h, k)", document.getElementById('vertexBadge'), { throwOnError: false });
      } else {
        document.getElementById('axisBadge').textContent = "x = -b / (2a)";
        document.getElementById('vertexBadge').textContent = "(h, k)";
      }
    });
  </script>
</body>
</html>

To find the vertex coordinates $(h, k)$, calculate $h = -\frac{b}{2a}$, then plug that $x$-value back into the function to find $k = f(h)$.

Finding the **$y$-intercept** is even faster: since $x = 0$ on the $y$-axis, $f(0) = a(0)^2 + b(0) + c = c$, giving the point $(0, c)$.

How do all of these steps work together on a real problem?

## Worked Example: Finding Every Key Feature

Let's find the orientation, axis of symmetry, vertex, and $y$-intercept for $f(x) = x^2 - 6x + 5$.

First, identify the coefficients: $a = 1$, $b = -6$, and $c = 5$. Since $a = 1 > 0$, the parabola opens upward, meaning the vertex is a minimum.

Next, calculate the axis of symmetry: $x = -\frac{-6}{2(1)} = \frac{6}{2} = 3$.

Now find the vertex $y$-value by substituting $x = 3$: $f(3) = (3)^2 - 6(3) + 5 = 9 - 18 + 5 = -4$, so the vertex is $(3, -4)$.

Finally, the constant term gives the $y$-intercept directly at $(0, 5)$.

<!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">
  <title>Key Features of a Quadratic Function</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      -webkit-tap-highlight-color: transparent;
    }
    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      padding: 12px;
      min-height: 100%;
    }
    .header {
      text-align: center;
      margin-bottom: 8px;
    }
    .equation {
      font-size: 18px;
      font-weight: 700;
      color: #0f172a;
      letter-spacing: -0.2px;
    }
    .nav-tabs {
      display: flex;
      flex-wrap: wrap;
      gap: 6px;
      justify-content: center;
      margin-bottom: 10px;
      width: 100%;
      max-width: 440px;
    }
    .tab-btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 20px;
      padding: 5px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      transition: all 0.25s ease;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
    }
    .tab-btn:hover {
      border-color: #cbd5e1;
      color: #1e2945;
    }
    .tab-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.35);
    }
    .graph-container {
      width: 100%;
      max-width: 400px;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e2e8f0;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      padding: 8px;
      position: relative;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    .axis-line {
      stroke: #cbd5e1;
      stroke-width: 1.5;
    }
    .grid-line {
      stroke: #f1f5f9;
      stroke-width: 1;
    }
    .tick-label {
      font-size: 10px;
      fill: #94a3b8;
      text-anchor: middle;
      font-weight: 500;
      user-select: none;
    }
    .parabola {
      fill: none;
      stroke: #0284c7;
      stroke-width: 3;
      stroke-linecap: round;
    }
    .sym-line {
      stroke: #8b5cf6;
      stroke-width: 2;
      stroke-dasharray: 5,4;
      transition: opacity 0.3s ease;
    }
    .feature-point {
      transition: transform 0.3s ease, opacity 0.3s ease;
      cursor: pointer;
    }
    .feature-point circle {
      stroke-width: 2;
    }
    .point-yint circle {
      fill: #10b981;
      stroke: #ffffff;
    }
    .point-vertex circle {
      fill: #ef4444;
      stroke: #ffffff;
    }
    .point-xint circle {
      fill: #f59e0b;
      stroke: #ffffff;
    }
    .label-badge {
      font-size: 11px;
      font-weight: 700;
      transition: opacity 0.3s ease;
      user-select: none;
    }
    .info-bar {
      margin-top: 8px;
      font-size: 13px;
      font-weight: 600;
      text-align: center;
      min-height: 22px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
    }
    .badge-pill {
      display: inline-block;
      padding: 2px 8px;
      border-radius: 6px;
      font-size: 12px;
    }
  </style>
</head>
<body>

  <div class="header">
    <div class="equation">f(x) = x² − 6x + 5</div>
  </div>

  <div class="nav-tabs">
    <button class="tab-btn active" data-step="0">All</button>
    <button class="tab-btn" data-step="1">y-Intercept</button>
    <button class="tab-btn" data-step="2">Axis of Symmetry</button>
    <button class="tab-btn" data-step="3">Vertex</button>
  </div>

  <div class="graph-container">
    <svg id="plot" viewBox="0 0 360 280">
      
      <g id="grid"></g>
      <g id="axes"></g>
      
      
      <line id="symLine" class="sym-line" x1="0" y1="0" x2="0" y2="0" opacity="0.9" />

      
      <path id="parabolaPath" class="parabola" d="" />

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

  <div class="info-bar" id="infoBar">
    <span class="badge-pill" style="background:#e0f2fe; color:#0369a1;">Tap tabs to explore key features</span>
  </div>

  <script>
    // Math coordinate system setup
    // x: [-1.5, 7.5], y: [-5.5, 6.5]
    const xMin = -1.5, xMax = 7.5;
    const yMin = -5.5, yMax = 6.5;
    const w = 360, h = 280;
    const pad = 24;

    function toScreenX(x) {
      return pad + ((x - xMin) / (xMax - xMin)) * (w - 2 * pad);
    }
    function toScreenY(y) {
      return h - (pad + ((y - yMin) / (yMax - yMin)) * (h - 2 * pad));
    }

    // Generate Grid & Axes
    const gridG = document.getElementById('grid');
    const axesG = document.getElementById('axes');

    for (let x = -1; x <= 7; x++) {
      const sx = toScreenX(x);
      gridG.innerHTML += `<line class="grid-line" x1="${sx}" y1="${pad}" x2="${sx}" y2="${h - pad}" />`;
      if (x !== 0) {
        axesG.innerHTML += `<text class="tick-label" x="${sx}" y="${toScreenY(0) + 12}">${x}</text>`;
      }
    }
    for (let y = -5; y <= 6; y++) {
      const sy = toScreenY(y);
      gridG.innerHTML += `<line class="grid-line" x1="${pad}" y1="${sy}" x2="${w - pad}" y2="${sy}" />`;
      if (y !== 0) {
        axesG.innerHTML += `<text class="tick-label" x="${toScreenX(0) - 8}" y="${sy + 3.5}">${y}</text>`;
      }
    }

    // Main Axes
    const ox = toScreenX(0);
    const oy = toScreenY(0);
    axesG.innerHTML += `<line class="axis-line" x1="${pad}" y1="${oy}" x2="${w - pad}" y2="${oy}" />`;
    axesG.innerHTML += `<line class="axis-line" x1="${ox}" y1="${pad}" x2="${ox}" y2="${h - pad}" />`;
    axesG.innerHTML += `<text class="tick-label" x="${ox - 7}" y="${oy + 11}">0</text>`;

    // Generate Parabola Path y = x^2 - 6x + 5
    const curvePoints = [];
    const xStart = 3 - Math.sqrt(4 + 6); // roughly -0.162 for y=6
    const xEnd = 3 + Math.sqrt(4 + 6);   // roughly 6.162 for y=6

    let dStr = '';
    for (let x = xStart; x <= xEnd; x += 0.05) {
      const y = x * x - 6 * x + 5;
      const px = toScreenX(x);
      const py = toScreenY(y);
      dStr += (dStr === '' ? 'M ' : ' L ') + px.toFixed(1) + ' ' + py.toFixed(1);
    }
    document.getElementById('parabolaPath').setAttribute('d', dStr);

    // Axis of Symmetry line
    const symX = toScreenX(3);
    const symLine = document.getElementById('symLine');
    symLine.setAttribute('x1', symX);
    symLine.setAttribute('y1', toScreenY(6.2));
    symLine.setAttribute('x2', symX);
    symLine.setAttribute('y2', toScreenY(-5.2));

    // Markers Definition
    const pointsData = [
      { id: 'yint', x: 0, y: 5, label: '(0, 5)', color: '#10b981', type: 'point-yint', lx: 14, ly: -6 },
      { id: 'vertex', x: 3, y: -4, label: 'Vertex (3, -4)', color: '#ef4444', type: 'point-vertex', lx: 0, ly: 16 },
      { id: 'xint1', x: 1, y: 0, label: '(1, 0)', color: '#f59e0b', type: 'point-xint', lx: -14, ly: -8 },
      { id: 'xint2', x: 5, y: 0, label: '(5, 0)', color: '#f59e0b', type: 'point-xint', lx: 14, ly: -8 }
    ];

    const pointsG = document.getElementById('points');
    pointsData.forEach(pt => {
      const px = toScreenX(pt.x);
      const py = toScreenY(pt.y);
      pointsG.innerHTML += `
        <g id="grp-${pt.id}" class="feature-point ${pt.type}">
          <circle cx="${px}" cy="${py}" r="5.5" />
          <text class="label-badge" x="${px + pt.lx}" y="${py + pt.ly}" fill="${pt.color}" text-anchor="${pt.lx === 0 ? 'middle' : (pt.lx > 0 ? 'start' : 'end')}">${pt.label}</text>
        </g>
      `;
    });

    // Step Info Content
    const stepInfo = [
      '<span class="badge-pill" style="background:#f1f5f9; color:#334155;">All Key Features</span>',
      '<span class="badge-pill" style="background:#ecfdf5; color:#047857;">y-Intercept: c = 5 → (0, 5)</span>',
      '<span class="badge-pill" style="background:#f5f3ff; color:#6d28d9;">Axis of Symmetry: x = −b / (2a) = 3</span>',
      '<span class="badge-pill" style="background:#fef2f2; color:#b91c1c;">Vertex: (3, f(3)) = (3, −4)</span>'
    ];

    const buttons = document.querySelectorAll('.tab-btn');
    const infoBar = document.getElementById('infoBar');

    function setStep(idx) {
      buttons.forEach((btn, i) => {
        btn.classList.toggle('active', i === idx);
      });

      infoBar.innerHTML = stepInfo[idx];

      const grpYint = document.getElementById('grp-yint');
      const grpVertex = document.getElementById('grp-vertex');
      const grpX1 = document.getElementById('grp-xint1');
      const grpX2 = document.getElementById('grp-xint2');

      if (idx === 0) { // All
        symLine.style.opacity = '0.9';
        grpYint.style.opacity = '1';
        grpVertex.style.opacity = '1';
        grpX1.style.opacity = '1';
        grpX2.style.opacity = '1';
      } else if (idx === 1) { // y-intercept
        symLine.style.opacity = '0.15';
        grpYint.style.opacity = '1';
        grpVertex.style.opacity = '0.15';
        grpX1.style.opacity = '0.15';
        grpX2.style.opacity = '0.15';
      } else if (idx === 2) { // Axis of symmetry
        symLine.style.opacity = '1';
        grpYint.style.opacity = '0.2';
        grpVertex.style.opacity = '0.4';
        grpX1.style.opacity = '0.2';
        grpX2.style.opacity = '0.2';
      } else if (idx === 3) { // Vertex
        symLine.style.opacity = '0.35';
        grpYint.style.opacity = '0.15';
        grpVertex.style.opacity = '1';
        grpX1.style.opacity = '0.15';
        grpX2.style.opacity = '0.15';
      }
    }

    buttons.forEach((btn, idx) => {
      btn.addEventListener('click', () => setStep(idx));
    });
  </script>
</body>
</html>

A common mistake is dropping the negative sign in the axis of symmetry formula when $b$ is already negative — remember that $-(-6)$ becomes positive $6$.

Try identifying $a$, $b$, and $c$ on your next equation first — once you have those three numbers, every landmark on the graph falls right into place.

math

Introduction to Quadratic Functions. Identify quadratic functions and their key features from equations and graphs.

Quadratic Functions & Equations: Intro to Parabolas and Key Features

Solving Quadratic Equations: Complex Solutions (HSN.CN.C.7)

Characteristics of Quadratic Functions: Graphs and Equations

To ensure mastery among 9th-10th grade students, bridge algebraic equations with visual representations dynamically. Start with concrete exploration of the leading coefficient $a$ and vertex $(h, k)$ using interactive tools like Desmos or GeoGebra. When introducing the zeros, emphasize the role of the discriminant $\Delta = b^2 - 4ac$ as a bridge connecting the geometric behavior of the parabola (whether it crosses, touches, or misses the $x$-axis) to algebraic number sets (two real, one real double, or two complex conjugate solutions). Scaffold the transition from real to complex roots by reinforcing $i = \sqrt{-1}$ and strictly practicing division across polynomial terms when simplifying the quadratic formula.

Introduction to Quadratic Functions

What Is a Quadratic Function?

The Anatomy of a Parabola