{
  "schemaVersion": "1.0",
  "moduleId": "calculus-functions-0580-pilot",
  "status": "prototype-scenarios",
  "pathways": {
    "differentiation": {
      "objectiveCode": "E2.12-1–4",
      "learnerAction": "Choose the correct derivative, then use it: gradient at a point, stationary point, or the turning point's y.",
      "objectiveVerbs": [
        "use derivatives of ax^n",
        "apply differentiation to gradients and stationary points",
        "discriminate between maxima and minima"
      ],
      "responseType": "model-construction",
      "scenarios": [
        {
          "id": "diff-mission-01",
          "prompt": "A curve has equation y = 3x^2 − 6x + 2. Choose the correct dy/dx on the board, then work out the gradient at x = 2.",
          "model": {
            "kind": "diff",
            "function": "3x^2 − 6x + 2",
            "terms": [
              [
                "3",
                2
              ],
              [
                "-6",
                1
              ],
              [
                "2",
                0
              ]
            ],
            "ask": "grad",
            "point": "2",
            "statX": null,
            "cardOptions": [
              "6x − 6",
              "3x − 6",
              "6x^2 − 6x",
              "6x − 4"
            ]
          },
          "expected": {
            "card": "6x − 6",
            "answer": "6"
          },
          "hint": "Multiply by the power, then take 1 off it — each term. The gradient at a point is dy/dx with x put in; a stationary point has dy/dx = 0, and the POINT needs its y too.",
          "misconceptionCode": "cf-power-not-multiplied"
        },
        {
          "id": "diff-mission-02",
          "prompt": "A curve has equation y = x^2 − 6x + 5. Choose the correct dy/dx on the board, then work out the x-coordinate of the stationary point.",
          "model": {
            "kind": "diff",
            "function": "x^2 − 6x + 5",
            "terms": [
              [
                "1",
                2
              ],
              [
                "-6",
                1
              ],
              [
                "5",
                0
              ]
            ],
            "ask": "stat",
            "point": null,
            "statX": "3",
            "cardOptions": [
              "2x − 6",
              "x − 6",
              "2x^2 − 6x",
              "2x − 1"
            ]
          },
          "expected": {
            "card": "2x − 6",
            "answer": "3"
          },
          "hint": "Multiply by the power, then take 1 off it — each term. The gradient at a point is dy/dx with x put in; a stationary point has dy/dx = 0, and the POINT needs its y too.",
          "misconceptionCode": "cf-point-needs-y"
        },
        {
          "id": "diff-mission-03",
          "prompt": "A curve has equation y = 2x^3 − 6x^2. Choose the correct dy/dx on the board, then work out the x-coordinate of the stationary point where x > 0.",
          "model": {
            "kind": "diff",
            "function": "2x^3 − 6x^2",
            "terms": [
              [
                "2",
                3
              ],
              [
                "-6",
                2
              ]
            ],
            "ask": "stat",
            "point": null,
            "statX": "2",
            "cardOptions": [
              "6x^2 − 12x",
              "2x^2 − 6x",
              "6x^3 − 12x^2",
              "−6x^2 + 12x"
            ]
          },
          "expected": {
            "card": "6x^2 − 12x",
            "answer": "2"
          },
          "hint": "Multiply by the power, then take 1 off it — each term. The gradient at a point is dy/dx with x put in; a stationary point has dy/dx = 0, and the POINT needs its y too.",
          "misconceptionCode": "cf-power-kept"
        },
        {
          "id": "diff-mission-04",
          "prompt": "A curve has equation y = −x^2 + 6x + 2. Choose the correct dy/dx on the board, then work out the y-coordinate of the turning point.",
          "model": {
            "kind": "diff",
            "function": "−x^2 + 6x + 2",
            "terms": [
              [
                "-1",
                2
              ],
              [
                "6",
                1
              ],
              [
                "2",
                0
              ]
            ],
            "ask": "maxmin",
            "point": null,
            "statX": "3",
            "cardOptions": [
              "−2x + 6",
              "−x + 6",
              "−2x^2 + 6x",
              "−2x + 8"
            ]
          },
          "expected": {
            "card": "−2x + 6",
            "answer": "11"
          },
          "hint": "Multiply by the power, then take 1 off it — each term. The gradient at a point is dy/dx with x put in; a stationary point has dy/dx = 0, and the POINT needs its y too.",
          "misconceptionCode": "cf-max-min-swapped"
        },
        {
          "id": "diff-mission-05",
          "prompt": "A curve has equation y = x^2 − 2x + 7. Choose the correct dy/dx on the board, then work out the y-coordinate of the turning point.",
          "model": {
            "kind": "diff",
            "function": "x^2 − 2x + 7",
            "terms": [
              [
                "1",
                2
              ],
              [
                "-2",
                1
              ],
              [
                "7",
                0
              ]
            ],
            "ask": "maxmin",
            "point": null,
            "statX": "1",
            "cardOptions": [
              "2x − 2",
              "x − 2",
              "2x^2 − 2x",
              "2x + 5"
            ]
          },
          "expected": {
            "card": "2x − 2",
            "answer": "6"
          },
          "hint": "Multiply by the power, then take 1 off it — each term. The gradient at a point is dy/dx with x put in; a stationary point has dy/dx = 0, and the POINT needs its y too.",
          "misconceptionCode": "cf-max-min-swapped"
        }
      ]
    },
    "functions": {
      "objectiveCode": "E2.13-1–3",
      "learnerAction": "Choose the correct inverse or composite expression, then evaluate it; read the range from the domain.",
      "objectiveVerbs": [
        "use function notation",
        "find inverse functions",
        "form composite functions"
      ],
      "responseType": "model-construction",
      "scenarios": [
        {
          "id": "func-mission-01",
          "prompt": "f(x) = 4x − 3. Choose the correct f⁻¹(x) on the board, then work out f⁻¹(5).",
          "model": {
            "kind": "func",
            "f": "4x − 3",
            "g": null,
            "ask": "inverse-value",
            "point": "5",
            "domain": null,
            "cardOptions": [
              "(x + 3) ÷ 4",
              "x ÷ 4 + 3",
              "¼x − 3",
              "(x − 3) ÷ 4"
            ]
          },
          "expected": {
            "card": "(x + 3) ÷ 4",
            "answer": "2"
          },
          "hint": "Undo the machine in reverse order for f⁻¹. For gf(x), f acts first, then g on f's output. Domain = inputs, range = outputs.",
          "misconceptionCode": "cf-order-not-reversed"
        },
        {
          "id": "func-mission-02",
          "prompt": "f(x) = 3x + 2 and g(x) = 2x − 1. Choose the correct gf(x) on the board, then work out gf(3).",
          "model": {
            "kind": "func",
            "f": "3x + 2",
            "g": "2x − 1",
            "ask": "composite-value",
            "point": "3",
            "domain": null,
            "cardOptions": [
              "6x + 3",
              "6x − 1",
              "(3x + 2)(2x − 1)",
              "6x + 5"
            ]
          },
          "expected": {
            "card": "6x + 3",
            "answer": "21"
          },
          "hint": "Undo the machine in reverse order for f⁻¹. For gf(x), f acts first, then g on f's output. Domain = inputs, range = outputs.",
          "misconceptionCode": "cf-composite-order"
        },
        {
          "id": "func-mission-03",
          "prompt": "f(x) = 2x − 7. Choose the correct f⁻¹(x) on the board, then work out f⁻¹(3).",
          "model": {
            "kind": "func",
            "f": "2x − 7",
            "g": null,
            "ask": "inverse-value",
            "point": "3",
            "domain": null,
            "cardOptions": [
              "(x + 7) ÷ 2",
              "x ÷ 2 + 7",
              "½x − 7",
              "(x − 7) ÷ 2"
            ]
          },
          "expected": {
            "card": "(x + 7) ÷ 2",
            "answer": "5"
          },
          "hint": "Undo the machine in reverse order for f⁻¹. For gf(x), f acts first, then g on f's output. Domain = inputs, range = outputs.",
          "misconceptionCode": "cf-order-not-reversed"
        },
        {
          "id": "func-mission-04",
          "prompt": "f(x) = 2x − 3 and g(x) = 4x + 1. Choose the correct gf(x) on the board, then work out gf(2).",
          "model": {
            "kind": "func",
            "f": "2x − 3",
            "g": "4x + 1",
            "ask": "composite-value",
            "point": "2",
            "domain": null,
            "cardOptions": [
              "8x − 11",
              "8x − 1",
              "(2x − 3)(4x + 1)",
              "8x − 13"
            ]
          },
          "expected": {
            "card": "8x − 11",
            "answer": "5"
          },
          "hint": "Undo the machine in reverse order for f⁻¹. For gf(x), f acts first, then g on f's output. Domain = inputs, range = outputs.",
          "misconceptionCode": "cf-composite-order"
        },
        {
          "id": "func-mission-05",
          "prompt": "f(x) = 3x + 2 with domain {0, 1, 2}. Choose what the RANGE means on the board, then work out the largest value in the range.",
          "model": {
            "kind": "func",
            "f": "3x + 2",
            "g": null,
            "ask": "domain",
            "point": null,
            "domain": [
              0,
              1,
              2
            ],
            "cardOptions": [
              "outputs of f on the domain",
              "the domain values themselves",
              "outputs of f⁻¹ on the domain",
              "inputs that make f(x) = 0"
            ]
          },
          "expected": {
            "card": "outputs of f on the domain",
            "answer": "8"
          },
          "hint": "Undo the machine in reverse order for f⁻¹. For gf(x), f acts first, then g on f's output. Domain = inputs, range = outputs.",
          "misconceptionCode": "cf-domain-range"
        }
      ]
    }
  }
}