Transcript: Understanding_the_graphs_of_quadratic_equations_20250703_110032.html

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  {
    "slide": 1,
    "fragments": [
      {
        "fragment_index": 1,
        "text_description": "Meet the Parabola",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Parabola",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "A parabola is the characteristic U-shaped graph of any quadratic function \\(y = ax^{2} + bx + c\\) with \\(a \\neq 0\\).\nQuick check: Which coefficient guarantees the curve is quadratic?",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 2,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Basic Upward Curve",
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      },
      {
        "fragment_index": 1,
        "text_description": "Graph of \\(y = x^{2}\\)",
        "image_description": "https://asset.sparkl.ac/pb/sparkl-vector-images/img_ncert/CSeGLPZ7MYUnljub8PO2sGNlNYUO6x223PpmX7RY.png"
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        "text_description": "Graph of \\(y = x^{2}\\)\nThe curve \\(y = x^{2}\\) is the base graph for all quadratic functions.\nIt opens upward, is symmetric about the y-axis, and its vertex—the lowest point—lies at the origin \\((0,0)\\).\nKey Points:\nBase graph for quadratic functions\nVertex at \\((0,0)\\)\nSymmetric about the y-axis",
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    ]
  },
  {
    "slide": 3,
    "fragments": []
  },
  {
    "slide": 4,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Opening Downward",
        "image_description": ""
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      {
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        "text_description": "Effect of Negative \\(a\\)\nWhen \\(a < 0\\), the parabola opens downward.\nThe graph is a reflection of the upward curve across the \\(x\\)-axis.\nIts vertex now marks the maximum value of the quadratic.\nKey Points:\nCoefficient \\(a < 0\\) ⇒ concave down.\nVertex becomes the maximum point.\nDownward graph mirrors the \\(a > 0\\) case.",
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  {
    "slide": 5,
    "fragments": [
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        "fragment_index": -1,
        "text_description": "Finding the Vertex",
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      {
        "fragment_index": 1,
        "text_description": "1\n\\[y = ax^2 + bx + c\\]\nStart with the general quadratic.",
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      },
      {
        "fragment_index": 2,
        "text_description": "2\n\\[y = a\\left(x^2 + \\frac{b}{a}x\\right) + c\\]\nFactor out \\(a\\) from the \\(x\\) terms.",
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      },
      {
        "fragment_index": 3,
        "text_description": "3\n\\[y = a\\left[\\left(x + \\frac{b}{2a}\\right)^2 - \\left(\\frac{b}{2a}\\right)^2\\right] + c\\]\nComplete the square inside the bracket.",
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      },
      {
        "fragment_index": 4,
        "text_description": "4\n\\[y = a\\left(x + \\frac{b}{2a}\\right)^2 - \\frac{b^{2}}{4a} + c\\]\nSimplify the constant term.",
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      },
      {
        "fragment_index": 5,
        "text_description": "5\n\\[Vertex\\ at\\ x = -\\frac{b}{2a}\\]\nSet the squared term to zero to get the x-coordinate.",
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      },
      {
        "fragment_index": 6,
        "text_description": "Key Insight:\nCompleting the square reveals the vertex formula \\(x = -\\frac{b}{2a}\\), essential for graphing any quadratic.",
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      }
    ]
  },
  {
    "slide": 6,
    "fragments": [
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        "fragment_index": -1,
        "text_description": "Shifted & Stretched",
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      },
      {
        "fragment_index": 1,
        "text_description": "Graph of \\(y = 2(x-1)^2 + 3\\)",
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      {
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        "text_description": "Translation & Dilation Explained\nCompare the parent curve \\(y = x^{2}\\) to \\(y = 2(x-1)^{2}+3\\).\nThe new coefficients show how far the graph slides and how much it stretches.\nKey Points:\nHorizontal translation: \\(x \\rightarrow x-1\\) moves the graph 1 unit right.\nVertical translation: \\(+3\\) lifts every point 3 units up.\nDilation: coefficient \\(a = 2\\) causes a vertical stretch, making the parabola narrower.\nResulting vertex: \\((1,\\,3)\\).",
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    ]
  },
  {
    "slide": 7,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Multiple Choice Question\nSubmit Answer\nCorrect!\nGreat! You identified both the correct vertex and the downward opening.\nIncorrect\nRecall: \\((h,k)\\) gives the vertex, and a negative \\(a\\) means the parabola opens downward.\nconst correctOption = 2;\n        const answerCards = document.querySelectorAll('.answer-card');\n        const submitBtn = document.getElementById('submitBtn');\n        const feedbackCorrect = document.getElementById('feedbackCorrect');\n        const feedbackIncorrect = document.getElementById('feedbackIncorrect');\n        \n        let selectedOption = null;\n        \n        answerCards.forEach((card, index) => {\n            card.addEventListener('click', () => {\n                answerCards.forEach(c => c.classList.remove('border-blue-500', 'bg-blue-50'));\n                card.classList.add('border-blue-500', 'bg-blue-50');\n                selectedOption = index;\n            });\n        });\n        \n        submitBtn.addEventListener('click', () => {\n            if (selectedOption === null) return;\n            \n            if (selectedOption === correctOption) {\n                feedbackCorrect.classList.remove('hidden');\n                feedbackIncorrect.classList.add('hidden');\n            } else {\n                feedbackIncorrect.classList.remove('hidden');\n                feedbackCorrect.classList.add('hidden');\n            }\n        });",
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      {
        "fragment_index": 1,
        "text_description": "Question\nFor \\(y = -3(x + 2)^2 - 4\\), which statement is true?",
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      },
      {
        "fragment_index": 2,
        "text_description": "1\nVertex at \\((-2,-4)\\) opens up.",
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      },
      {
        "fragment_index": 3,
        "text_description": "2\nVertex at \\((2,-4)\\) opens down.",
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      },
      {
        "fragment_index": 4,
        "text_description": "3\nVertex at \\((-2,-4)\\) opens down.",
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      },
      {
        "fragment_index": 5,
        "text_description": "4\nVertex at \\((2,4)\\) opens up.",
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      },
      {
        "fragment_index": 6,
        "text_description": "Hint:\nIn \\(y = a(x - h)^2 + k\\), the vertex is \\((h,k)\\). If \\(a<0\\), the parabola opens downward.",
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      }
    ]
  },
  {
    "slide": 8,
    "fragments": [
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        "fragment_index": -1,
        "text_description": "Key Takeaways\nThank You!\nWe hope you found this lesson informative and engaging.",
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      },
      {
        "fragment_index": 1,
        "text_description": "Recap: sign of \\(a\\) decides opening—up for \\(a>0\\), down for \\(a<0\\).",
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      {
        "fragment_index": 2,
        "text_description": "\\(b\\) slides the parabola left or right, while \\(c\\) moves it up or down.",
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      },
      {
        "fragment_index": 3,
        "text_description": "Vertex at \\(x = -\\frac{b}{2a}\\) marks the curve’s peak or valley.",
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      },
      {
        "fragment_index": 4,
        "text_description": "Graph is symmetric about the vertical line through the vertex.",
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      }
    ]
  }
]