Vertex Form Calculator

Work out vertex form instantly with clear inputs, formula shown and shareable results.

About Vertex Form Calculator

Completing the square rewrites ax² + bx + c as a(x − h)² + k with h = −b/(2a) and k = c − b²/(4a). The vertex is then read straight off, and the axis of symmetry is x = h.

Formulas

Vertex

h = −b / (2a), k = c − b² / (4a)

Vertex form

y = a(x − h)² + k

Worked example

With the default values below, Vertex Form Calculator returns Vertex form of y = 2(x - 3)^2 + 5. Change any input above and every figure here is recalculated in your browser.

Inputs used

Coefficient a: 2
Coefficient b: -12
Coefficient c: 23

Results

Vertex form: y = 2(x - 3)^2 + 5
Vertex x (axis of symmetry): 3.00000000
Vertex y: 5.00000000
Direction: Opens upward, vertex is the minimum

Step by step

1Values used
2Vertex
3Vertex form
4Vertex form= y = 2(x - 3)^2 + 5
5Vertex x (axis of symmetry)= 3.00000000
6Vertex y= 5.00000000
7Direction= Opens upward, vertex is the minimum

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For y = 2x² − 12x + 23?

h = 3 and k = 23 − 144/8 = 5, so y = 2(x − 3)² + 5 with a minimum at (3, 5).

How does the sign of a matter?

Positive a opens the parabola upward so the vertex is the lowest point; negative a opens it downward so the vertex is the highest.