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Parabola Vertex Calculator

This parabola vertex calculator finds the vertex from the standard form f(x) = ax² + bx + c – or from vertex form. For x² − 4x + 3 the vertex is at (2 | −1); you also get the vertex form, the roots, the axis of symmetry and the working.

Example: You have: Standard form f(x) = ax² + bx + c · a 1 · b -4 · c 3 → Vertex: Vertex (2 | −1). Source: Wikipedia – Parabel (Mathematik): Scheitelpunkt, Scheitelform. Updated: .

Coefficient of the square, not 0. Positive: opens upward
Coefficient of x
Constant term
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Result

Vertex
Vertex (2 | −1)
Vertex form
f(x) = (x − 2)² − 1
Standard form
f(x) = x² − 4x + 3
Roots
Two roots: x₁ = 1 and x₂ = 3
Properties
  • Opens upward (a > 0): the vertex is the minimum.
  • Axis of symmetry: x = 2
  • y-intercept: (0 | 3)
  • Range: y ≥ −1
Working
  • Vertex x: x_v = −b ÷ (2a) = −(−4) ÷ (2 · 1) = 2
  • Vertex y: substituting x_v into f gives y_v = −1.
  • Completing the square: f(x) = 1·(x − 2)² + −1, because 3 − 1·2² = −1.

How it is calculated

The vertex is the lowest or highest point of a parabola – where it turns around. For f(x) = ax² + bx + c it lies at x = −b ÷ (2a); the y-value comes from substituting. In vertex form f(x) = a(x − h)² + k you can read it off directly: (h | k).

How the tool works

Example

f(x) = x² − 4x + 3: x_v = 4 ÷ 2 = 2, y_v = 3 − 16 ÷ 4 = −1. The vertex form is f(x) = (x − 2)² − 1 and the roots are 1 and 3. In reverse, the vertex form f(x) = 2(x − 3)² − 5 expands to 2x² − 12x + 13. If you only want the roots, use the quadratic formula calculator; the curve itself is drawn by the graphing calculator.

Frequently asked questions

How do you find the vertex of a parabola?

Use x_v = −b ÷ (2a) and substitute x_v into the function for y_v. For f(x) = x² − 4x + 3 the vertex is (2 | −1).

What is vertex form?

f(x) = a(x − h)² + k. The vertex is (h | k): h shifts the parabola to the right, k upward.

How do I convert standard form to vertex form?

By completing the square: ax² + bx + c = a(x + b ÷ 2a)² + c − b² ÷ 4a. The tool shows the steps with your numbers.

How can I tell whether the vertex is a maximum or a minimum?

From the sign of a: for a > 0 the parabola opens upward and the vertex is a minimum, for a < 0 it is a maximum.

What does “no real roots” mean?

The parabola lies entirely above or below the x-axis and never touches it. The vertex is then on the same side as the parabola.

Sources and legal basis

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