Math & learning

Vertex form calculator

Turn quadratic coefficients into vertex form and locate the turning point.

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  • Calculated on your device

01 / Your quadratic

Start with y = ax² + bx + c. Enter the signed number for each coefficient; use 0 when the x or constant term is absent.

Must be nonzero.

In a(x − h)² + k, the vertex is (h, k). A positive a opens upward; a negative a opens downward.

Enter decimals or scientific notation, rather than expressions such as 1/2. Each coefficient must be between −1 billion and 1 billion; |a| must be at least 0.000000001.

02 / Your results

(x − 2)² − 1Vertex form · y ≈ a(x − h)² + k
(2, -1)Vertex (h, k)Opens up · minimum at x ≈ 2
Key features of your parabola
FeatureValue
Axis of symmetryx ≈ 2
Y-intercept(0, 3)
Minimum y-value≈ -1
  1. h = −b / (2a) = −(-4) / (2 × 1) ≈ 2
  2. k = c − b² / (4a) = (3) − (-4)² / (4 × 1) ≈ -1
  3. Substitute a, h and k into y = a(x − h)² + k: y ≈ (x − 2)² − 1

Values are rounded for display. The ≈ sign means approximately equal; this calculator uses floating-point arithmetic, not exact fractions.

How the calculation works

For y = ax² + bx + c with a ≠ 0, calculate h = −b ÷ (2a), then k = ah² + bh + c. The same quadratic is y = a(x − h)² + k, with vertex (h, k). Decimal outputs are numerical approximations when rounding is needed.

Try an example

Enter a = 1, b = −4 and c = 3. Then h = 2 and k = −1, giving y = (x − 2)² − 1 and vertex (2, −1).

How to use this calculator

  1. Write your quadratic as y = ax² + bx + c. Enter the numerical values of a, b and c, keeping their signs; enter 0 for a missing x term or constant.
  2. Select Convert to vertex form. Read the expression a(x − h)² + k and the vertex coordinates (h, k).
  3. Check the steps h = −b ÷ (2a) and k = ah² + bh + c. The axis of symmetry is x = h; positive a opens upward and negative a opens downward.

Common questions

Why can a not be zero?

When a is zero, the x² term disappears. The expression is linear or constant, so it has no quadratic vertex form and the calculation would divide by zero.

Why is the sign inside the brackets different from h?

Vertex form uses x − h. If h is 2, the bracket is x − 2; if h is −2, it is x + 2. The vertex keeps the coordinate h itself.

Does this provide exact fractions or symbolic algebra?

No. Enter numerical coefficients in the three fields, not an equation or a fraction such as 1/3. Results use numerical arithmetic and may be rounded. An approximate decimal is not an exact symbolic fraction.

Further reading: OpenStax: quadratic functions and vertex form