{
  "schema_version": 1,
  "body": {
    "type": "doc",
    "blocks": [
      {
        "type": "heading",
        "level": 2,
        "content": [{ "type": "text", "text": "What makes a quadratic?" }]
      },
      {
        "type": "paragraph",
        "content": [
          { "type": "text", "text": "A quadratic function has the standard form " },
          { "type": "math", "latex": "y = ax^2 + bx + c" },
          { "type": "text", "text": " with " },
          { "type": "math", "latex": "a \\neq 0" },
          {
            "type": "text",
            "text": ". Its graph is a parabola: it opens upward when "
          },
          { "type": "math", "latex": "a > 0" },
          { "type": "text", "text": " and downward when " },
          { "type": "math", "latex": "a < 0" },
          { "type": "text", "text": "." }
        ]
      },
      {
        "type": "heading",
        "level": 2,
        "content": [{ "type": "text", "text": "The vertex" }]
      },
      {
        "type": "paragraph",
        "content": [
          {
            "type": "text",
            "text": "The turning point of the parabola is called the vertex. Completing the square rewrites the standard form as"
          }
        ]
      },
      {
        "type": "math_block",
        "latex": "y = a\\left(x + \\frac{b}{2a}\\right)^2 + c - \\frac{b^2}{4a},"
      },
      {
        "type": "paragraph",
        "content": [
          { "type": "text", "text": "so the vertex sits at " },
          {
            "type": "math",
            "latex": "\\left(-\\frac{b}{2a},\\; c - \\frac{b^2}{4a}\\right)"
          },
          {
            "type": "text",
            "text": ". The squared term is never negative, which is exactly why the function is extremal there."
          }
        ]
      },
      {
        "type": "callout",
        "variant": "tip",
        "blocks": [
          {
            "type": "paragraph",
            "content": [
              {
                "type": "text",
                "text": "You rarely need the full formula for the y-coordinate. Find x = -b/2a first, then substitute — it is faster and less error-prone."
              }
            ]
          }
        ]
      },
      {
        "type": "heading",
        "level": 2,
        "content": [{ "type": "text", "text": "Why physicists care" }]
      },
      {
        "type": "paragraph",
        "content": [
          {
            "type": "text",
            "text": "Constant-acceleration motion is quadratic in time, so projectile trajectories are parabolas. The vertex of the height-versus-time graph is the peak of the flight — you will use this in the problem set."
          }
        ]
      }
    ]
  },
  "problems": ["quadratic-vertex"]
}