Vertex Form Calculator

Vertex form is y = a(x − h)² + k, where (h, k) is the vertex of the parabola. It makes the turning point of the parabola immediately visible without any additional calculation. The value of 'a' determines whether the parabola opens upward (a > 0) or downward (a < 0).

To convert y = ax² + bx + c to vertex form, you complete the square or use the vertex formulas: h = −b / (2a) and k = c − b² / (4a). Then substitute those values into y = a(x − h)² + k. This calculator does all of that automatically when you enter a, b, and c.

Given y = ax² + bx + c, the x-coordinate of the vertex is h = −b / (2a), and the y-coordinate is k = c − b² / (4a). You can also find k by substituting h back into the original equation: k = a·h² + b·h + c.

To go from y = a(x − h)² + k to standard form, expand the squared term: a(x − h)² = a(x² − 2hx + h²) = ax² − 2ahx + ah². Then add k to get y = ax² − 2ahx + (ah² + k), where b = −2ah and c = ah² + k.

Vertex form is especially useful when you need to quickly identify the maximum or minimum value of a quadratic function (the vertex), graph a parabola, or solve optimization problems. It is also used in physics for projectile motion equations where the peak height corresponds to the vertex.

The vertex is the turning point of the parabola — the highest point if the parabola opens downward (a < 0), or the lowest point if it opens upward (a > 0). In real-world applications, the vertex often represents a maximum profit, minimum cost, or peak height in a trajectory.

No. If a = 0, the equation becomes linear (y = bx + c) rather than quadratic, and there is no parabola or vertex. The calculator requires a ≠ 0 to produce a valid result.

If the vertex is at (2, 5) and the leading coefficient is a, the vertex form is y = a(x − 2)² + 5. You still need to know 'a' to fully define the parabola. If you also know the standard form coefficients, enter them above to find both h and k automatically.