Vertex Form Calculator

Convert a quadratic equation from standard form to vertex form and find its vertex, axis of symmetry, direction, and minimum or maximum value.

Convert to Vertex Form

Standard Form

ax² + bx + c = 0

Coefficient of x²

Cannot be zero.

Coefficient of x

Enter the x coefficient.

Constant

Enter the constant.

Examples:

Vertex Form

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Calculation Steps

What Is Vertex Form?

Vertex form is a way of writing a quadratic function that makes the vertex easy to identify.

y = a(x − h)² + k

In this form, the vertex of the parabola is (h, k). The value of a determines whether the parabola opens upward or downward.

How to Use This Calculator

Enter the coefficients from your quadratic equation in standard form.

Enter the coefficient a.
Enter the coefficient b.
Enter the constant c.
Click the calculation button.
Review the vertex form and graph information.

How to Convert Standard Form to Vertex Form

A quadratic equation can be written in standard form as:

y = ax² + bx + c

The vertex can be found using the x-coordinate formula:

h = −b / 2a

After finding h, substitute it into the original quadratic function to calculate k. The vertex is therefore (h, k).

The resulting equation can then be expressed as:

y = a(x − h)² + k

Vertex of a Parabola

The vertex is the highest or lowest point of a quadratic graph.

If a > 0, the parabola opens upward and the vertex represents its minimum value.

If a < 0, the parabola opens downward and the vertex represents its maximum value.

Axis of Symmetry

Every parabola has a vertical line of symmetry that passes through its vertex.

x = h = −b / 2a

This line divides the parabola into two matching halves.

Example: Convert a Quadratic to Vertex Form

Consider the quadratic:

y = x² − 6x + 5

Here, a = 1, b = −6, and c = 5.

First calculate the x-coordinate of the vertex:

h = −(−6) / (2 × 1) = 3

Substitute x = 3 into the equation:

k = 3² − 6(3) + 5 = −4

Therefore, the vertex is (3, −4), and the vertex form is:

y = (x − 3)² − 4

Frequently Asked Questions

What is vertex form?

Vertex form is y = a(x − h)² + k. It directly shows the vertex of a quadratic as (h, k).

How do I find the vertex of a quadratic?

For y = ax² + bx + c, calculate h = −b/(2a). Then substitute h into the original equation to find k. The vertex is (h, k).

What does the value of a tell us?

If a is positive, the parabola opens upward. If a is negative, it opens downward.

What is the axis of symmetry?

The axis of symmetry is the vertical line passing through the vertex. Its equation is x = h.

Can this calculator handle negative coefficients?

Yes. You can enter positive, negative, decimal, and fractional-equivalent decimal coefficients.

What happens if a is zero?

If a is zero, the equation is linear rather than quadratic, so vertex form for a parabola does not apply.