Transcript: Understanding_graphs_of_quadratic_equations_20250701_122338.html

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[
  {
    "slide": 1,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Quadratic Equation – General Form",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "\\(ax^{2}+bx+c=0\\)",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "A quadratic equation can always be written as \\(ax^{2}+bx+c=0\\) with real numbers \\(a, b, c\\) and \\(a\\neq0\\). In this form \\(a\\) is the quadratic coefficient, \\(b\\) is the linear coefficient, and \\(c\\) is the constant term.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 2,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "The Parabola: Key Features",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "Diagram of a parabola with vertex, axis of symmetry and arms labeled",
        "image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/presentation_images/asset.sparkl.me/pb/presentation/1256/images/729e647f970a4cba150b44eb131e821d.png"
      },
      {
        "fragment_index": 2,
        "text_description": "What to spot on the graph\nThe graph of any quadratic equation forms a U-shaped curve called a parabola.\nFind the vertex at the top or bottom. The vertical line through it is the axis of symmetry. The arms pointing up or down show the opening direction.",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Key Points:\nShape: U-shaped parabola\nVertex: highest or lowest point\nAxis of symmetry: vertical line \\(x = h\\)\nOpening direction: arms up if \\(a > 0\\), down if \\(a < 0\\)",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 3,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Effect of Coefficient ‘a’ on Opening Direction",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "Positive \\(a\\)\n\\(a > 0\\)\nParabola opens upward — looks like a smile.",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Negative \\(a\\)\n\\(a < 0\\)\nParabola opens downward — looks like a frown.",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Key Similarity\nSign of \\(a\\) alone predicts whether the parabola opens up or down.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 4,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Coefficient ‘a’ and the Width of the Parabola",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "Wide (blue), normal (green) and narrow (purple) parabolas",
        "image_description": "/static/images/parabola-width.png"
      },
      {
        "fragment_index": 2,
        "text_description": "Effect of |a| on Width\nThe absolute value \\(|a|\\) alone decides how wide or steep a parabola is.\nLarger \\(|a|\\) stretches the graph vertically, making it steeper and narrower.\nSmaller \\(|a|\\) compresses it, producing a flatter, wider curve.\nKey Points:\n\\(|a| > 1\\): narrow, steep (vertical stretch)\n\\(|a| < 1\\): wide, flat (vertical compression)\nWidth depends only on \\(|a|\\), not on the sign of \\(a\\).",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 5,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Role of Coefficient ‘b’ – Moving the Axis",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "Parabolas before and after changing \\(b\\)",
        "image_description": "/static/images/parabola_shift.png"
      },
      {
        "fragment_index": 2,
        "text_description": "Horizontal shift & completing the square\nChanging \\(b\\) shifts the entire parabola left or right.\nThe axis of symmetry is \\(x=-\\frac{b}{2a}\\).\nCompleting the square confirms the vertex lies on this axis.\nKey Points:\n\\(b>0\\): axis moves left \\(\\frac{b}{2a}\\) units.\n\\(b<0\\): axis moves right \\(\\frac{|b|}{2a}\\) units.\nShift leaves opening direction and width set by \\(a\\) unchanged.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 6,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Role of Coefficient ‘c’ – The y-Intercept\nConstant term = y-intercept",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "Same curve shifted up or down as \\(c\\) varies",
        "image_description": "https://sparkl-vector-images.s3.ap-south-1.amazonaws.com/img/lp/study_content/editlive_lp/80/2013_07_09_11_22_19/fboxarld_6120782470407155088.png"
      },
      {
        "fragment_index": 2,
        "text_description": "For \\(y = ax^{2} + bx + c\\), when \\(x = 0\\) we get \\(y = c\\).",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "So the parabola crosses the y-axis at \\((0, c)\\); changing \\(c\\) slides it vertically.",
        "image_description": ""
      },
      {
        "fragment_index": 4,
        "text_description": "Key Points:\nConstant term \\(c\\) equals the y-intercept.\n\\(c > 0\\): graph shifts up  \\(c < 0\\): graph shifts down.\nShape and width stay unchanged—only vertical position moves.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 7,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Match Equation to Graph\nCheck Answers\nResults\n// Drag and drop functionality\n        const draggableItems = document.querySelectorAll('.draggable-item');\n        const dropZones = document.querySelectorAll('.drop-zone');\n        const checkAnswersBtn = document.getElementById('checkAnswersBtn');\n        const feedbackArea = document.getElementById('feedbackArea');\n        const feedbackContent = document.getElementById('feedbackContent');\n\n        draggableItems.forEach(item => {\n            item.addEventListener('dragstart', handleDragStart);\n            item.addEventListener('dragend', handleDragEnd);\n        });\n\n        dropZones.forEach(zone => {\n            zone.addEventListener('dragover', handleDragOver);\n            zone.addEventListener('drop', handleDrop);\n            zone.addEventListener('dragenter', handleDragEnter);\n            zone.addEventListener('dragleave', handleDragLeave);\n        });\n\n        function handleDragStart(e) {\n            e.target.classList.add('opacity-50');\n            e.dataTransfer.setData('text/plain', e.target.dataset.id);\n        }\n\n        function handleDragEnd(e) {\n            e.target.classList.remove('opacity-50');\n        }\n\n        function handleDragOver(e) {\n            e.preventDefault();\n        }\n\n        function handleDragEnter(e) {\n            e.preventDefault();\n            e.target.closest('.drop-zone').classList.add('border-green-500', 'bg-green-50');\n        }\n\n        function handleDragLeave(e) {\n            e.target.closest('.drop-zone').classList.remove('border-green-500', 'bg-green-50');\n        }\n\n        function handleDrop(e) {\n            e.preventDefault();\n            const dropZone = e.target.closest('.drop-zone');\n            dropZone.classList.remove('border-green-500', 'bg-green-50');\n\n            const itemId = e.dataTransfer.getData('text/plain');\n            const draggedItem = document.querySelector(`[data-id=\"${itemId}\"]`);\n\n            if (draggedItem && dropZone) {\n                dropZone.appendChild(draggedItem);\n                dropZone.querySelector('.text-center').style.display = 'none';\n            }\n        }\n\n        // Check answers functionality\n        checkAnswersBtn.addEventListener('click', () => {\n            feedbackArea.classList.remove('hidden');\n            feedbackContent.innerHTML = '<p class=\"text-green-600\">Answers checked! Review your results above.</p>';\n        });",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "Drag each quadratic equation to the parabola that matches its form. This reinforces form-to-graph recognition.",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Draggable Items\n\\(\\,y = x^{2}\\, \\)\n\\(\\,y = (x-2)^{2} + 1\\,\\)\n\\(\\,y = -x^{2}+4\\,\\)",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Drop Zones\nGraph A\nGraph B\nGraph C",
        "image_description": ""
      },
      {
        "fragment_index": 4,
        "text_description": "Tip:\nCheck the vertex and whether the parabola opens up or down to spot the correct match.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 8,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Zeros, Roots & the Discriminant\nApplications",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "\\[\\Delta = b^{2} - 4ac\\]",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Variable Definitions\na\ncoefficient of \\(x^{2}\\)\nb\ncoefficient of \\(x\\)\nc\nconstant term\n\\(\\Delta\\)\ndiscriminant",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "\\( \\Delta > 0 \\)\nTwo real roots — parabola crosses x-axis twice.",
        "image_description": ""
      },
      {
        "fragment_index": 4,
        "text_description": "\\( \\Delta = 0 \\)\nOne real root — parabola touches x-axis.",
        "image_description": ""
      },
      {
        "fragment_index": 5,
        "text_description": "\\( \\Delta < 0 \\)\nNo real roots — parabola misses x-axis.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 9,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Key Takeaways\nGraphs of Quadratic Equations",
        "image_description": ""
      },
      {
        "fragment_index": 1,
        "text_description": "Opening Direction\nSign of \\(a\\): \\(a>0\\) opens up, \\(a<0\\) opens down.",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Parabola Width\nLarger \\(|a|\\) → narrower curve; smaller \\(|a|\\) → wider curve.",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Horizontal Position\nCoefficient \\(b\\) moves vertex sideways; axis of symmetry \\(x=-\\frac{b}{2a}\\).",
        "image_description": ""
      },
      {
        "fragment_index": 4,
        "text_description": "Vertical Position\nConstant \\(c\\) shifts graph up or down; gives the \\(y\\)-intercept.",
        "image_description": ""
      },
      {
        "fragment_index": 5,
        "text_description": "Number of Roots\nDiscriminant \\(b^{2}-4ac\\): >0 two roots, =0 one root, <0 none.",
        "image_description": ""
      }
    ]
  }
]