{
  "id": "vertex",
  "version": "26717418820f",
  "status": "published",
  "name": "Vertex Calculator",
  "question": "What is the vertex of my parabola?",
  "summary": "Computes the vertex (h, k) of a parabola from y = ax² + bx + c or y = a(x − h)² + k, with the axis of symmetry, the minimum or maximum, the y-intercept, the real roots and both forms.",
  "category": "math",
  "subcategory": "algebra",
  "url": "https://www.acalculator.org/math/vertex-calculator",
  "markdown": "https://www.acalculator.org/math/vertex-calculator.md",
  "kind": "function",
  "method": "h = −b ÷ (2a), k = c − b² ÷ (4a); from vertex form, b = −2ah and c = ah² + k; roots x = h ± √(−k ÷ a).",
  "assumptions": [
    "a may not be 0 (then the graph is a line with no vertex).",
    "Coefficients are read as the exact decimals typed, so the vertex is an exact fraction."
  ],
  "inputs": {
    "$schema": "https://json-schema.org/draft/2020-12/schema",
    "type": "object",
    "properties": {
      "form": {
        "title": "My equation is",
        "description": "Standard form y = ax² + bx + c, or vertex form y = a(x − h)² + k.",
        "type": "string",
        "enum": [
          "standard",
          "vertex"
        ]
      },
      "a": {
        "title": "a",
        "description": "The coefficient of x² (or of the square in vertex form). It may not be 0.",
        "type": "number",
        "minimum": -1000000,
        "maximum": 1000000
      },
      "b": {
        "title": "b",
        "description": "The coefficient of x.",
        "type": "number",
        "minimum": -1000000,
        "maximum": 1000000
      },
      "c": {
        "title": "c",
        "description": "The constant term.",
        "type": "number",
        "minimum": -1000000,
        "maximum": 1000000
      },
      "h": {
        "title": "h",
        "description": "The x of the vertex in a(x − h)² + k.",
        "type": "number",
        "minimum": -1000000,
        "maximum": 1000000
      },
      "k": {
        "title": "k",
        "description": "The y of the vertex in a(x − h)² + k.",
        "type": "number",
        "minimum": -1000000,
        "maximum": 1000000
      }
    }
  },
  "outputs": {
    "vertex": {
      "label": "Vertex (h, k)",
      "description": "The turning point of the parabola, as exact fractions.",
      "format": "text"
    },
    "h": {
      "label": "h (x of the vertex)",
      "description": "h = −b ÷ (2a), as a decimal.",
      "format": "number"
    },
    "k": {
      "label": "k (y of the vertex)",
      "description": "k = c − b² ÷ (4a), as a decimal.",
      "format": "number"
    },
    "axis": {
      "label": "Axis of symmetry",
      "description": "The vertical line through the vertex, x = h.",
      "format": "text"
    },
    "opens": {
      "label": "Opens",
      "description": "Up with a minimum at the vertex when a > 0; down with a maximum when a < 0.",
      "format": "text"
    },
    "standard": {
      "label": "Standard form",
      "description": "y = ax² + bx + c.",
      "format": "text"
    },
    "vertexForm": {
      "label": "Vertex form",
      "description": "y = a(x − h)² + k.",
      "format": "text"
    },
    "b": {
      "label": "b",
      "description": "The coefficient of x: b = −2ah.",
      "format": "number"
    },
    "c": {
      "label": "c (y-intercept)",
      "description": "The constant term c = ah² + k, where the parabola crosses the y-axis.",
      "format": "number"
    },
    "roots": {
      "label": "Real roots (x-intercepts)",
      "description": "x = h ± √(−k ÷ a), when −k ÷ a ≥ 0.",
      "format": "text"
    }
  },
  "defaultAnswer": {
    "inputs": {
      "form": "standard",
      "a": 2,
      "b": -6,
      "c": 7,
      "h": 1.5,
      "k": 2.5
    },
    "outputs": {
      "vertex": "(3/2, 5/2)",
      "h": 1.5,
      "k": 2.5,
      "axis": "x = 3/2",
      "opens": "Up: the minimum is y = 5/2",
      "standard": "y = 2x² − 6x + 7",
      "vertexForm": "y = 2(x − 3/2)² + 5/2",
      "b": -6,
      "c": 7,
      "roots": "None: the parabola does not cross the x-axis."
    },
    "text": "The vertex is (3/2, 5/2), on the axis x = 3/2."
  },
  "examples": [
    {
      "given": {
        "form": "standard",
        "a": 2,
        "b": -6,
        "c": 7
      },
      "expect": {
        "vertex": "(3/2, 5/2)",
        "h": 1.5,
        "k": 2.5,
        "vertexForm": "y = 2(x − 3/2)² + 5/2",
        "opens": "Up: the minimum is y = 5/2"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions, Example 3 (vertex (3/2, 5/2); f(x) = 2(x − 3/2)² + 5/2), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions"
    },
    {
      "given": {
        "form": "standard",
        "a": 1,
        "b": -4,
        "c": 3
      },
      "expect": {
        "vertex": "(2, −1)",
        "axis": "x = 2",
        "roots": "x = 1 and x = 3",
        "c": 3
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions (h = −b ÷ (2a), k = f(h)), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions; hand calculation in content.mdx: h = 4 ÷ 2 = 2, k = 4 − 8 + 3 = −1, x = 2 ± 1"
    },
    {
      "given": {
        "form": "vertex",
        "a": -0.5,
        "h": -2,
        "k": 8
      },
      "expect": {
        "standard": "y = −(1/2)x² − 2x + 6",
        "b": -2,
        "c": 6,
        "opens": "Down: the maximum is y = 8",
        "roots": "x = −6 and x = 2"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions (f(x) = a(x − h)² + k; a < 0 opens downward), https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions; hand calculation in content.mdx: b = −2 × (−0.5) × (−2) = −2, c = −0.5 × 4 + 8 = 6"
    },
    {
      "given": {
        "form": "standard",
        "a": 1,
        "b": 2,
        "c": -1
      },
      "expect": {
        "vertex": "(−1, −2)",
        "roots": "x = −2.414213562 and x = 0.4142135624"
      },
      "source": "OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions, https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions; hand calculation in content.mdx: h = −1, k = 1 − 2 − 1 = −2, x = −1 ± √2"
    }
  ],
  "sources": [
    "OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions (vertex h = −b/(2a), k = f(h); standard form f(x) = a(x − h)² + k; axis of symmetry x = −b/(2a); a > 0 opens upward; Example 3: f(x) = 2x² − 6x + 7 has vertex (3/2, 5/2)). https://openstax.org/books/algebra-and-trigonometry-2e/pages/5-1-quadratic-functions (retrieved 2026-10-02)"
  ],
  "related": [
    "quadratic-formula",
    "slope",
    "factoring",
    "polynomial"
  ],
  "changelog": []
}