Find the vertex and axis of symmetry of a parabola from standard, vertex or factored form — with the vertex form of the equation worked out for you.
Standard form: y = ax² + bx + c.
Vertex (h, k)
—How to use this tool
Standard (ax² + bx + c), vertex (a(x − h)² + k) or factored (a(x − r₁)(x − r₂)).
Labels change with the form, so you always know which number goes where.
The turning point (h, k) is shown at the top, with the axis of symmetry and the vertex form of the equation.
A positive a opens upwards (the vertex is a minimum); a negative a opens downwards (the vertex is a maximum).
Formula & worked example
y = x² − 4x + 3: h = −(−4) ÷ 2 = 2, k = 4 − 8 + 3 = −1, so the vertex is (2, −1), the axis is x = 2, and the vertex form is y = (x − 2)² − 1.
Limitations — read this before you rely on it
This finds the vertex of a parabola only. If a = 0 the equation is a straight line with no vertex, and the calculator says so instead of inventing an answer. Complex (non-real) roots are not needed for the vertex.
Frequently asked questions
For y = ax² + bx + c, the vertex x-coordinate is −b ÷ 2a; substitute it back to get the y-coordinate.
The vertical line x = h through the vertex. The parabola mirrors across it, so points at equal distances either side have equal y values.
The axis is midway between the roots: (r₁ + r₂) ÷ 2. Substitute that x back into the equation for the vertex y value.
If a is positive the parabola opens upwards and the vertex is the minimum; if a is negative it opens downwards and the vertex is the maximum.
y = a(x − h)² + k, where (h, k) is the vertex. It makes the turning point readable at a glance.