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Quadratic Vertex Calculator

Find the vertex, axis of symmetry, minimum or maximum and vertex form of a parabola y = ax² + bx + c.

a (x² coefficient)

b (x coefficient)

c (constant)

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How it works

h = −b ÷ 2a · k = c − b² ÷ 4a · axis: x = h · vertex form: y = a(x − h)² + k

Every parabola y = ax² + bx + c has a single turning point, the vertex, sitting on its axis of symmetry at x = −b ÷ 2a. Substituting that x back into the equation gives the vertex’s y-value, which simplifies to k = c − b² ÷ 4a. The sign of a decides direction: positive opens upward so the vertex is the lowest point (a minimum), negative opens downward so the vertex is the highest point (a maximum). Writing the parabola as y = a(x − h)² + k puts that vertex on display.

Worked example

For y = x² − 4x + 3 (a = 1, b = −4, c = 3): the axis is x = −(−4) ÷ (2·1) = 2. Then k = 3 − (−4)² ÷ 4 = 3 − 4 = −1, so the vertex is (2, −1). Because a = 1 is positive the parabola opens upward, making −1 the minimum value, and the vertex form is y = (x − 2)² − 1.

Sources: Quadratic function — Wikipedia

Frequently asked questions

How do I know if the vertex is a minimum or a maximum?

Look at a. When a is positive the parabola opens upward and the vertex is the lowest point, a minimum. When a is negative it opens downward and the vertex is the highest point, a maximum.

What is the axis of symmetry?

It is the vertical line x = −b ÷ 2a passing through the vertex. The parabola is a mirror image across this line, so two points at equal horizontal distance from it share the same height.

How does vertex form relate to standard form?

Vertex form y = a(x − h)² + k and standard form y = ax² + bx + c describe the same parabola. Vertex form shows the turning point (h, k) directly, while standard form shows the y-intercept c. Expanding one gives the other.

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This calculator is for educational purposes. Double-check important results before acting on them.