Vertex Form Calculator — Standard to Vertex Form Converter
Enter a, b and c of y = ax² + bx + c to get the vertex form y = a(x − h)² + k, the vertex (h, k), axis of symmetry, y-intercept, real roots and a live parabola plot.
The axis of symmetry passes through x = h
- 1
−b
−(-4) = 4 - 2
2a
2 × 1 = 2 - 3
Vertex x-coordinate h = −b ÷ (2a)
4 ÷ 2 = 2h is the axis of symmetry; the vertex lies directly above or below this x-value.
How does this calculator work?
Convert y = ax² + bx + c to vertex form by computing h = −b/(2a) and k = c − b²/(4a), giving y = a(x − h)² + k. The vertex is (h, k) and is the turning point. Real x-intercepts exist when b² − 4ac ≥ 0 and are found via x = (−b ± √(b²−4ac)) / (2a).
Formula
How this is calculated
Any quadratic y = ax² + bx + c can be rewritten in vertex form y = a(x − h)² + k by completing the square. The vertex (h, k) is the turning point of the parabola: h = −b / (2a) gives the x-coordinate of the axis of symmetry, and k = c − b² / (4a) is the corresponding y-value — the minimum when a > 0 and the maximum when a < 0.
The discriminant D = b² − 4ac determines the number of real roots: D > 0 gives two distinct x-intercepts found via x = (−b ± √D) / (2a); D = 0 gives one repeated root at x = h; and D < 0 means the parabola never crosses the x-axis. The y-intercept is always at (0, c).
All intermediate values are computed in full floating-point precision. The parabola is plotted at 61 evenly spaced points centred on the vertex so the shape is accurate regardless of the coefficient magnitudes. The vertex V is marked in red on the plot; any real roots appear at the x-axis.
Frequently asked questions
Vertex form y = a(x − h)² + k places the turning point (h, k) directly in the equation, so you can read the maximum or minimum value and the axis of symmetry without further calculation. It is especially useful for graphing parabolas and for optimisation problems where you need the extreme value.
Use h = −b / (2a) and k = c − b² / (4a), then write y = a(x − h)² + k. Alternatively, factor a out of the x-terms, add and subtract (b/2a)² inside the parentheses to complete the square, and simplify the constant terms.
A negative discriminant (b² − 4ac < 0) means the parabola never crosses the x-axis — there are no real roots. In vertex form this corresponds to k and a having the same sign: the vertex sits entirely above (a > 0, k > 0) or below (a < 0, k < 0) the x-axis.
Also known as
TG we-Calculate Editorial Team. (2026). Vertex Form Calculator — Standard to Vertex Form Converter [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/vertex-form-calculator
TG we-Calculate Editorial Team. "Vertex Form Calculator — Standard to Vertex Form Converter." TG we-Calculate. 2026. https://we-calculate.com/calculator/vertex-form-calculator.
TG we-Calculate Editorial Team, "Vertex Form Calculator — Standard to Vertex Form Converter," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/vertex-form-calculator
@misc{wecalculate_vertex_form_calculator, title = {Vertex Form Calculator — Standard to Vertex Form Converter}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/vertex-form-calculator}}, year = {2026}, note = {TG we-Calculate} }
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