Vertex Calculator
Find the vertex of a parabola from its quadratic equation in standard form (ax² + bx + c). Also calculates the axis of symmetry and direction.
Enter coefficients for y = ax² + bx + c
Enter coefficients to find the vertex.
Vertex Formula
h = −b / (2a)
k = f(h) = a·h² + b·h + c
Vertex = (h, k)
k = f(h) = a·h² + b·h + c
Vertex = (h, k)
The vertex is the highest or lowest point on a parabola, depending on whether the parabola opens down or up.
How to Find the Vertex
- Identify a, b, and c from the standard form y = ax² + bx + c.
- Calculate h = −b / (2a). This is the x-coordinate of the vertex.
- Substitute h back into the equation to find k (the y-coordinate).
Example: y = 2x² − 8x + 6
h = −(−8) / (2 × 2) = 8 / 4 = 2
k = 2(2)² − 8(2) + 6 = 8 − 16 + 6 = −2
Vertex = (2, −2)
h = −(−8) / (2 × 2) = 8 / 4 = 2
k = 2(2)² − 8(2) + 6 = 8 − 16 + 6 = −2
Vertex = (2, −2)
Vertex Form of a Quadratic
y = a(x − h)² + k
Once you know the vertex (h, k), you can rewrite the equation in vertex form. This form makes it easy to identify the vertex directly.
FAQ
How do I know if the vertex is a maximum or minimum?
If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, it opens downward and the vertex is a maximum.
What is the axis of symmetry?
The axis of symmetry is the vertical line x = h that passes through the vertex. Every parabola is symmetric about this line.
Related Calculators
Last updated: 2026-08-08