Transcript: Understanding_the_graphs_of_quadratic_equations_20250703_104802.html

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  {
    "slide": 1,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "What is a Parabola?",
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      },
      {
        "fragment_index": 1,
        "text_description": "Parabola",
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      },
      {
        "fragment_index": 2,
        "text_description": "A parabola is a smooth, U-shaped curve produced by the graph of any quadratic equation \\(y = ax^{2} + bx + c\\).",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Key Characteristics:\nSymmetric about its vertex.\nOpens up when \\(a > 0\\); down when \\(a < 0\\).\nRepresents every quadratic equation.",
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      {
        "fragment_index": 4,
        "text_description": "Example:\nThe curve of \\(y = x^{2}\\) is an upward-opening parabola.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 2,
    "fragments": [
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        "fragment_index": -1,
        "text_description": "General Quadratic Form",
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      },
      {
        "fragment_index": 1,
        "text_description": "\\[y = ax^2 + bx + c\\]\nIdentify \\(a\\), \\(b\\), \\(c\\) to predict how the parabola looks.",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Variable Definitions\na\nOpens up/down & sets width\nb\nMoves axis of symmetry\nc\nY-intercept (vertical shift)\nx\nIndependent variable\ny\nDependent output value",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Applications\nQuick Sketch\nUse signs of \\(a, b, c\\) to foresee opening and intercept before drawing.\nPhysics Trajectories\nChanging coefficients alters a projectile’s height, range, and symmetry.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 3,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Base Shape y = x²",
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      },
      {
        "fragment_index": 1,
        "text_description": "Graph of \\(y = x^{2}\\)",
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        "text_description": "Parent Quadratic Graph\nThe parent curve \\(y = x^{2}\\) forms a smooth U-shape called a parabola.\nIt is perfectly symmetric about the y-axis.\nThe lowest point, or vertex, is at the origin \\((0,0)\\).",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Key Points:\nOpens upward, creating a U-shape.\nAxis of symmetry: \\(x = 0\\).\nVertex at \\((0,0)\\) is the minimum.\nPasses through \\((\\pm1,1)\\).",
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      }
    ]
  },
  {
    "slide": 4,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Changing 'a' − Stretch & Flip",
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      {
        "fragment_index": 1,
        "text_description": "Parabolas for varying values of \\(a\\)",
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      },
      {
        "fragment_index": 2,
        "text_description": "Effect of coefficient \\(a\\)\nMagnitude decides the width; sign decides the opening.\nKey Points:\n\\(|a| > 1\\): Narrow, vertically stretched.\n\\(0 < |a| < 1\\): Wide, vertically shrunk.\n\\(a < 0\\): Parabola reflects and opens downward.",
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      }
    ]
  },
  {
    "slide": 5,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Upward vs Downward",
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      {
        "fragment_index": 1,
        "text_description": "Opens Up (a > 0)\nSign of \\(a\\) is positive.\nU-shaped; arms rise as \\(|x|\\) increases.\nVertex gives the minimum \\(y\\)-value.",
        "image_description": ""
      },
      {
        "fragment_index": 2,
        "text_description": "Opens Down (a < 0)\nSign of \\(a\\) is negative.\n∩-shaped; arms fall as \\(|x|\\) increases.\nVertex gives the maximum \\(y\\)-value.",
        "image_description": ""
      },
      {
        "fragment_index": 3,
        "text_description": "Key Similarities\nBoth are parabolas of \\(y = ax^{2} + bx + c\\).\nAxis of symmetry at \\(x = -\\frac{b}{2a}\\).\nVertex is the graph’s extreme point.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 6,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Role of 'c' − Y-Intercept",
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        "text_description": "",
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      },
      {
        "fragment_index": 2,
        "text_description": "Constant Term ‘c’",
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      },
      {
        "fragment_index": 3,
        "text_description": "In \\(y = ax^2 + bx + c\\), the constant term sets the graph’s vertical position.",
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      },
      {
        "fragment_index": 4,
        "text_description": "It equals the y-value when \\(x = 0\\); this fixes the y-intercept.",
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      },
      {
        "fragment_index": 5,
        "text_description": "Key Points:\nIncrease c → graph slides up; width and direction unchanged.\nDecrease c → graph slides down; same shape remains.\nOnly the y-intercept and vertex height shift by c.",
        "image_description": ""
      }
    ]
  },
  {
    "slide": 7,
    "fragments": [
      {
        "fragment_index": -1,
        "text_description": "Quick Check\nCorrect!\nGreat work! You correctly applied \\(-\\frac{b}{2a}\\).\nIncorrect\nCheck \\(-\\frac{b}{2a}\\) again with \\(a = 1\\) and \\(b = 4\\).\nconst correctOption = 1;\n        const answerCards = document.querySelectorAll('.answer-card');\n        const submitBtn = document.getElementById('slide-09-g3v91k-submit');\n        const feedbackCorrect = document.getElementById('slide-09-g3v91k-feedback-correct');\n        const feedbackIncorrect = document.getElementById('slide-09-g3v91k-feedback-incorrect');\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 the parabola \\(y = x^{2} + 4x + 1\\), what is the x–coordinate of its axis of symmetry?",
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      },
      {
        "fragment_index": 2,
        "text_description": "1\n\\(x = -4\\)",
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      },
      {
        "fragment_index": 3,
        "text_description": "2\n\\(x = -2\\)",
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      },
      {
        "fragment_index": 4,
        "text_description": "3\n\\(x = 2\\)",
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      },
      {
        "fragment_index": 5,
        "text_description": "4\n\\(x = 4\\)",
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      },
      {
        "fragment_index": 6,
        "text_description": "Hint:\nUse the formula \\(-\\frac{b}{2a}\\) to locate the axis of symmetry.",
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      },
      {
        "fragment_index": 7,
        "text_description": "Submit Answer",
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      }
    ]
  },
  {
    "slide": 8,
    "fragments": [
      {
        "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": "Coefficient \\(a\\): sign flips the opening; larger \\(|a|\\) narrows, smaller widens.",
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      },
      {
        "fragment_index": 2,
        "text_description": "Coefficient \\(b\\): shifts the vertex sideways, giving the graph its horizontal position.",
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      },
      {
        "fragment_index": 3,
        "text_description": "Coefficient \\(c\\): raises or lowers the whole parabola; the y-intercept is \\((0,c)\\).",
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      },
      {
        "fragment_index": 4,
        "text_description": "Master these three moves to sketch any quadratic quickly and accurately.",
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      }
    ]
  }
]