Vertex Form Calculator

Standard form to vertex form

y = ax² + bx + c → y = a(x − h)² + k h = −b ÷ 2a, k = c − b² ÷ 4a

The vertex (h, k) is the highest or lowest point of the parabola. If a is positive the parabola opens upward; if negative it opens downward.

Worked example

For y = 2x² − 8x + 6: h = 8 ÷ 4 = 2 and k = 6 − 64 ÷ 8 = −2, so the vertex form is y = 2(x − 2)² − 2. The discriminant is 64 − 48 = 16, giving roots x = 1 and x = 3.

What the discriminant tells you

b² − 4ac	Roots
Positive	Two real roots
Zero	One repeated real root
Negative	Two complex roots

Frequently asked questions

How do I find the vertex of a parabola?

Use x = −b ÷ 2a for the x-coordinate, then substitute it into the equation to get y.

What is the quadratic formula?

x = (−b ± √(b² − 4ac)) ÷ 2a. It gives the roots of any quadratic equation.

What is completing the square?

A method of rewriting ax² + bx + c as a(x − h)² + k, which is exactly the vertex form this calculator produces.

Can a quadratic have no x-intercepts?

Yes. If the discriminant is negative, the parabola never crosses the x-axis.