Converts a quadratic equation from standard form (y = ax² + bx + c) into vertex form (y = a(x − h)² + k), revealing the parabola’s vertex directly.
How it works
Completing the square on the standard-form equation gives the vertex coordinates h and k, which are then substituted into the vertex-form template.
What this does not include
This does not include finding the roots (x-intercepts) of the quadratic — this calculator specifically converts between forms and identifies the vertex.
How to use this calculator
- Enter the coefficients a, b, and c from the standard-form equation.
A worked example
The quadratic y = x² − 4x + 3 converts to vertex form y = (x − 2)² − 1 — vertex at (2, −1).
y = 2x² + 8x + 5 converts with vertex at h = −2, k = −3.
What the variables mean
| Variable | Meaning |
|---|---|
| a, b, c | Coefficients of the standard form y = ax² + bx + c |
| h, k | The vertex coordinates in vertex form y = a(x − h)² + k |
Edge cases worth knowing
An a coefficient of zero means the equation isn’t actually a parabola — it collapses to a straight line, which has no vertex in the parabolic sense, so the calculator declines to show a result.
Vertex form directly reveals the parabola’s maximum or minimum point — reading (h, k) straight off the equation, without needing calculus or a graph.
Frequently asked questions
Why is vertex form useful?
It immediately shows the parabola’s vertex (its highest or lowest point) without any additional calculation, which standard form doesn’t reveal directly.
What does it mean if a is zero?
The equation isn’t actually a quadratic — without an x² term, it’s a linear equation instead, which doesn’t have a parabola vertex to find.
Does the sign of a tell me anything about the parabola?
Yes — a positive a means the parabola opens upward (the vertex is a minimum); a negative a means it opens downward (the vertex is a maximum).
