Vertex Form Calculator (standard form ↔ vertex form of a parabola)

Understanding the vertex form calculator — concept and principle

This calculator converts a quadratic between standard form y = ax² + bx + c and vertex form y = a(x − h)² + k. It gives the vertex, the axis of symmetry, whether the parabola opens up or down, the minimum or maximum value, the x-intercepts and the y-intercept. Use it for algebra homework, graphing parabolas and optimisation problems.

h = −b ÷ 2a = 2 and k = y(2) = −5, so the lowest point of the parabola is (2, −5).

How it is calculated

Standard to vertex: h = −b ÷ (2a), k = c − b² ÷ (4a).
Vertex to standard: b = −2ah, c = ah² + k.
Axis of symmetry: x = h.
a > 0 opens upward (k is the minimum); a < 0 opens downward (k is the maximum).
x-intercepts: h ± √(−k/a) when −k/a ≥ 0. y-intercept: c.

The principle

Vertex form comes from completing the square. Factor a out of the x terms, then add and subtract the square of half the x-coefficient: ax² + bx + c = a(x² + (b/a)x) + c = a(x + b/(2a))² + c − b²/(4a). Because a squared term is never negative, a(x − h)² is smallest (when a > 0) or largest (when a < 0) at x = h, so the vertex is (h, k). The parabola is symmetric about the vertical line through the vertex.

Worked example

Default inputs: standard form, a = 2, b = −8, c = 3.

h = −(−8) ÷ (2 × 2) = 2.
k = 3 − (−8)² ÷ (4 × 2) = 3 − 64 ÷ 8 = −5.
Vertex form: y = 2(x − 2)² − 5; vertex (2, −5); axis x = 2.
a = 2 > 0, so it opens upward and −5 is the minimum.
x-intercepts: 2 ± √(5/2) ≈ 0.4188611699 and 3.58113883; y-intercept 3.

How to read the result

The vertex gives the lowest or highest point, which is the answer to many optimisation questions (maximum height, minimum cost). If k and a have the same sign, the parabola never crosses the x-axis and there are no real x-intercepts. A larger |a| makes the parabola narrower.

Common mistakes

Sign error in h: y = a(x − h)², so y = 2(x − 2)² has h = +2, while y = 2(x + 2)² has h = −2.
Forgetting to multiply by a when expanding back.
Completing the square without first factoring out a.
Reading k as the y-intercept; the y-intercept is c.

Keep in mind: The calculator works with numeric coefficients only. Check rounded intercepts by substituting them back into the equation.

Frequently asked questions

How do I find the vertex of y = ax² + bx + c?

h = −b/(2a), then k = the value of y at x = h.

What does a tell you in vertex form?

The direction (up if positive) and how wide or narrow the parabola is.

Why is vertex form useful?

It shows the vertex and axis directly, making the graph and the minimum or maximum easy to read.

Sources: OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions

Checked 9 Nov 2026. OpenStax, Algebra and Trigonometry 2e, §5.1 Quadratic Functions (standard form f(x) = a(x − h)² + k, vertex h = −b/(2a), k = f(h)); OpenStax, Algebra 1, §7.15.2 The Vertex Form; Intermediate Algebra 2e, §9.2 for completing the square.

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