{
  "$schema": "../../../schemas/mko-v0.4.schema.json",
  "schema_version": "mko-v0.4",
  "id": "mko-quadratic-function",
  "semantic_family_id": "sf-quadratic-function",
  "type": "definition",
  "version": "0.10.0",
  "language": "zh-Hant",
  "titles": {
    "zh-Hant": "頂點式二次函數"
  },
  "summary": {
    "zh-Hant": "實數全論域的 q(x)=a(x−h)²+k、a≠0；頂點、全域極值與對稱點的明示切片。"
  },
  "statement": {
    "zh-Hant": "令 q:R→R，q(x)=a(x−h)²+k，其中 a、h、k 為實數且 a≠0。其頂點為 (h,k)，對稱軸為 x=h；a>0 時全域最小值為 k，a<0 時全域最大值為 k，均在 x=h 取得。圖上 (x,q(x)) 關於該軸的對稱點為 (2h−x,q(x))。"
  },
  "formula": {
    "tex": "q(x)",
    "mathml": "<math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math>",
    "semantic_ast": {
      "type": "function_call",
      "callee": {
        "type": "symbol",
        "name": "q",
        "glyph": "q"
      },
      "arguments": [
        {
          "type": "symbol",
          "name": "x",
          "glyph": "x"
        }
      ]
    },
    "compiler": {
      "id": "ocme-formula-core",
      "version": "0.4.0",
      "source_sha256": "2e8e41ba989d33241c9ca3f2eee9a081e18fe18e753cd7f2b8f3ade796031d07"
    }
  },
  "assumptions": [
    {
      "id": "a1",
      "zh-Hant": "自變數遍及全部實數；若改成限制區間，極值不能不加檢查直接套用。"
    },
    {
      "id": "a2",
      "zh-Hant": "a≠0，否則不是本條目的二次函數；題庫參數限定整數，使用精確代入。"
    },
    {
      "id": "a3",
      "zh-Hant": "代表公式 q(x) 只表示函數求值；完整頂點式與條件在敘述中。"
    }
  ],
  "symbols": [
    {
      "token": "q",
      "role": "quadratic_real_function",
      "role_zh": "實數頂點式二次函數",
      "scope": "representative_formula"
    },
    {
      "token": "x",
      "role": "real_input",
      "role_zh": "實數輸入",
      "scope": "representative_formula"
    }
  ],
  "dependencies": [
    {
      "id": "mko-function-mapping",
      "title_zh": "函數映射",
      "reason_zh": "宣告實數總函數的輸入與輸出。"
    },
    {
      "id": "mko-quadratic-equation",
      "title_zh": "二次方程式",
      "reason_zh": "二次項與非零係數的語義。"
    }
  ],
  "explanation": {
    "paragraphs_zh": [
      "平方非負，所以正 a 向上開口、負 a 向下開口。頂點是座標對，不是單一極值；另一方向沒有全域最大或最小值。",
      "代入時先計 x−h，再平方、乘 a、加 k。負係數仍保持二次項，不能把負 a 改取絕對值。",
      "x 與 2h−x 距離軸相同，平方項相等。x=h 時對稱點就是自身；移動 h 與輸入 x 同量不構成新的代入計算。"
    ],
    "common_misconception_zh": "把頂點與極值混用，或在限制區間、零係數下直接沿用全實數二次結論。"
  },
  "proofs": [],
  "computational_companions": [],
  "verification": {
    "computational_status": "not_evaluated_no_computational_claim",
    "warning_zh": "整數頂點式切片可精確重算；全域極值及圖形對稱尚未附形式 Evidence。",
    "evidence_refs": [],
    "producers": [
      {
        "id": "lean-mathlib",
        "adapter": "formal_system",
        "config_path": "formal/lean/lakefile.toml",
        "status": "configured"
      }
    ]
  },
  "formalization": {
    "status": "not_formalized",
    "target_systems": [
      "Lean 4",
      "Mathlib"
    ],
    "next_obligation_zh": "綁定平方非負、完整實數論域及頂點式恆等式；不把二次方程式 proof 冒充本物件 proof。"
  },
  "provenance": {
    "origin": "ocme_local_ai_materialization_candidate",
    "note_zh": "依 research/atlas-quadratic-function.md 整理；2026-10-05 脈衝 R02（使用者手動接續 2026-10-06 工作序列；實際日期仍為 2026-10-05），英文及 Lean 依使用者指示延後。",
    "content_license": "CC-BY-SA-4.0",
    "code_license": "MIT",
    "created_at": "2026-10-05"
  }
}