{
  "api_version": "v1",
  "publisher": {
    "name": "Manazoo",
    "url": "https://manazoo.com/"
  },
  "license": "All rights reserved（AI の参照・要約と出典リンクつきの引用は可。転載・再配布・販売は不可）",
  "terms_url": "https://manazoo.com/terms/",
  "disclaimer": "内容の正確さには気をつけていますが、すべてが正しいとは限りません。テストや受験の前には、教科書や先生の説明でも確かめてください。間違いを見つけたら意見箱から知らせてください。",
  "attribution": "事実データの一部は Wikidata（CC0）による。単元の構成は文部科学省「中学校学習指導要領」「高等学校学習指導要領」（政府標準利用規約2.0）。",
  "kind": "quiz",
  "unit": {
    "slug": "quadratic-functions",
    "title": "二次関数",
    "url": "https://manazoo.com/high/math-1/quadratic-functions/"
  },
  "canonical_url": "https://manazoo.com/high/math-1/quadratic-functions/quiz/",
  "questions": [
    {
      "id": "q-cs-axis-value",
      "type": "input",
      "prompt": "y＝2x²−12x＋7の軸を表す直線はx＝□である。□に当てはまる数を求めよ。",
      "choices": null,
      "answer": "3",
      "accept": [],
      "explanation": "y＝2(x²−6x)＋7＝2{(x−3)²−9}＋7＝2(x−3)²−11。軸はx＝3。",
      "entities": [
        "completing-the-square-and-vertex"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cs-correct-form-pick",
      "type": "choice",
      "prompt": "y＝3x²＋12x＋7を平方完成した式として正しいものはどれか。",
      "choices": [
        "3(x＋2)²−5",
        "3(x＋4)²−5",
        "3(x＋2)²＋3",
        "3(x−2)²−5"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "3x²＋12x＋7＝3(x²＋4x)＋7＝3{(x＋2)²−4}＋7＝3(x＋2)²−12＋7＝3(x＋2)²−5。「3(x＋2)²＋3」は( )の中の−4に3をかけ忘れた間違い。",
      "entities": [
        "completing-the-square-and-vertex"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cs-negative-a-vertex",
      "type": "choice",
      "prompt": "y＝−2x²＋4x＋1を平方完成して求めた頂点はどれか。",
      "choices": [
        "(1, −3)",
        "(2, 3)",
        "(−1, 3)",
        "(1, 3)"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "y＝−2(x²−2x)＋1＝−2{(x−1)²−1}＋1＝−2(x−1)²＋2＋1＝−2(x−1)²＋3。頂点は(1, 3)。",
      "entities": [
        "completing-the-square-and-vertex"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-cs-vertex-y-coord",
      "type": "input",
      "prompt": "y＝x²−6x＋10を平方完成すると、頂点の座標は(3, k)になる。kの値を求めよ。",
      "choices": null,
      "answer": "1",
      "accept": [],
      "explanation": "x²−6x＋10＝(x−3)²−9＋10＝(x−3)²＋1。頂点は(3, 1)なのでk＝1。",
      "entities": [
        "completing-the-square-and-vertex"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-mm-domain-restricted-max",
      "type": "choice",
      "prompt": "y＝x²−4x＋6（1≤x≤5）の最大値を求めよ。",
      "choices": [
        "2",
        "3",
        "6",
        "11"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "y＝(x−2)²＋2。頂点はx＝2で範囲内だが最小値2を与える。端ではx＝1のとき3、x＝5のとき(5−2)²＋2＝11。最大値は11。",
      "entities": [
        "maximum-and-minimum-of-quadratic-functions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-mm-perimeter24-max-area",
      "type": "input",
      "prompt": "周の長さが24cmの長方形で、面積が最大になるときの面積は何cm²か。",
      "choices": null,
      "answer": "36",
      "accept": [],
      "explanation": "1辺をxcmとすると、もう1辺は(12−x)cm。面積y＝x(12−x)＝−(x−6)²＋36。x＝6で最大値36をとる。",
      "entities": [
        "maximum-and-minimum-of-quadratic-functions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-mm-simple-minimum",
      "type": "input",
      "prompt": "y＝(x−2)²＋3の最小値を求めよ。",
      "choices": null,
      "answer": "3",
      "accept": [],
      "explanation": "(x−2)²は0以上の値しかとらないので、x＝2のとき(x−2)²＝0で最小。y＝0＋3＝3が最小値。",
      "entities": [
        "maximum-and-minimum-of-quadratic-functions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-mm-vertex-outside-domain-min",
      "type": "choice",
      "prompt": "y＝(x−5)²−2（0≤x≤3）の最小値はどれか。",
      "choices": [
        "−2",
        "2",
        "23",
        "7"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "頂点x＝5は範囲0≤x≤3の外。この範囲では値が減り続けるので、最小は右端x＝3：(3−5)²−2＝2。",
      "entities": [
        "maximum-and-minimum-of-quadratic-functions"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-qgt-axis-and-direction",
      "type": "choice",
      "prompt": "二次関数y＝−3(x＋2)²−4のグラフの軸と、開く向きを正しく述べたものはどれか。",
      "choices": [
        "軸はx＝4で、下に開く",
        "軸はx＝2で、下に開く",
        "軸はx＝−2で、上に開く",
        "軸はx＝−2で、下に開く"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "y＝−3(x−(−2))²−4の形なので軸はx＝−2。a＝−3＜0なので下に開く。",
      "entities": [
        "quadratic-function-graph-and-translation"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-qgt-translate-right4-down1",
      "type": "input",
      "prompt": "y＝x²を右に4、下に1平行移動すると、y＝(x−4)²＋qの形になる。qの値を求めよ。",
      "choices": null,
      "answer": "-1",
      "accept": [
        "−1",
        "－1"
      ],
      "explanation": "右に4動かすのでxをx−4に置き換え、下に1動かすので最後に1を引く。y＝(x−4)²−1となり、q＝−1。",
      "entities": [
        "quadratic-function-graph-and-translation"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-qgt-vertex-from-equation",
      "type": "choice",
      "prompt": "y＝2(x−3)²＋5のグラフの頂点はどれか。",
      "choices": [
        "(3, −5)",
        "(5, 3)",
        "(3, 5)",
        "(−3, 5)"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "y＝a(x−p)²＋qの頂点は(p, q)。p＝3、q＝5なので頂点は(3, 5)。",
      "entities": [
        "quadratic-function-graph-and-translation"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-qgt-vertex-to-translation",
      "type": "choice",
      "prompt": "y＝x²のグラフを平行移動して、頂点が(−2, 6)になるようにした。平行移動の向きを正しく述べたものはどれか。",
      "choices": [
        "右に2、下に6",
        "左に2、上に6",
        "左に2、下に6",
        "右に2、上に6"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "頂点が(0, 0)から(−2, 6)へ動くので、xは−2（左に2）、yは＋6（上に6）動いている。",
      "entities": [
        "quadratic-function-graph-and-translation"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-qi-always-positive",
      "type": "choice",
      "prompt": "x²＋2x＋5＞0を解け。",
      "choices": [
        "x＞−1",
        "−1＜x＜5",
        "すべての実数",
        "解なし"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "D＝2²−4×1×5＝−16＜0なので、グラフはx軸と交わらない。a＝1＞0なのでグラフは常に0より上。解はすべての実数。",
      "entities": [
        "quadratic-inequalities"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-qi-no-solution",
      "type": "choice",
      "prompt": "−x²＋2x−5＞0を解け。",
      "choices": [
        "x＝1",
        "x＞1",
        "すべての実数",
        "解なし"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "D＝2²−4×(−1)×(−5)＝4−20＝−16＜0なので、グラフはx軸と交わらない。a＝−1＜0なのでグラフは常に0より下。＞0を満たすxは無い。",
      "entities": [
        "quadratic-inequalities"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-qi-solve-negative-between",
      "type": "choice",
      "prompt": "x²−6x＋8＜0を解け。",
      "choices": [
        "x＜2またはx＞4",
        "すべての実数",
        "2＜x＜4",
        "x＜2"
      ],
      "answer": 2,
      "accept": [],
      "explanation": "零点はx＝2, 4。a＝1＞0なので、グラフは零点の内側で0より下。解は2＜x＜4。",
      "entities": [
        "quadratic-inequalities"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-qi-solve-positive-outside",
      "type": "choice",
      "prompt": "x²−6x＋8＞0を解け。",
      "choices": [
        "x＜4またはx＞8",
        "x＜2またはx＞4",
        "x＞4",
        "2＜x＜4"
      ],
      "answer": 1,
      "accept": [],
      "explanation": "x²−6x＋8＝0の解はx＝2, 4。a＝1＞0なので、グラフは零点の外側で0より上。解はx＜2またはx＞4。",
      "entities": [
        "quadratic-inequalities"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-xi-discriminant-and-count",
      "type": "choice",
      "prompt": "y＝x²−2x−3の判別式Dの値と、グラフとx軸の共有点の個数を正しく組み合わせたものはどれか。",
      "choices": [
        "D＝16、共有点0個",
        "D＝4、共有点1個",
        "D＝−16、共有点2個",
        "D＝16、共有点2個"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "D＝(−2)²−4×1×(−3)＝4＋12＝16。D＞0なので共有点は2個。",
      "entities": [
        "quadratic-function-and-x-axis-intersections"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-xi-discriminant-zero",
      "type": "input",
      "prompt": "y＝x²−6x＋9の判別式Dの値を求めよ。",
      "choices": null,
      "answer": "0",
      "accept": [],
      "explanation": "D＝(−6)²−4×1×9＝36−36＝0。グラフはx軸に接し、共有点は1個。",
      "entities": [
        "quadratic-function-and-x-axis-intersections"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-xi-negative-discriminant-count",
      "type": "choice",
      "prompt": "y＝2x²＋3x＋5のグラフとx軸の共有点は何個か。",
      "choices": [
        "0個",
        "1個",
        "2個",
        "3個"
      ],
      "answer": 0,
      "accept": [],
      "explanation": "D＝3²−4×2×5＝9−40＝−31。D＜0なので共有点は0個。",
      "entities": [
        "quadratic-function-and-x-axis-intersections"
      ],
      "verified": true,
      "placeholder": false
    },
    {
      "id": "q-xi-roots-from-equation",
      "type": "choice",
      "prompt": "y＝x²−7x＋10のグラフとx軸の共有点の座標はどれか。",
      "choices": [
        "(2, 0)のみ",
        "(−2, 0)と(−5, 0)",
        "(1, 0)と(10, 0)",
        "(2, 0)と(5, 0)"
      ],
      "answer": 3,
      "accept": [],
      "explanation": "x²−7x＋10＝0を因数分解すると(x−2)(x−5)＝0。x＝2, 5なので共有点は(2, 0)と(5, 0)。",
      "entities": [
        "quadratic-function-and-x-axis-intersections"
      ],
      "verified": true,
      "placeholder": false
    }
  ]
}