What this calculator does
Vertex form rewrites a quadratic as a(x − h)² + k instead of ax² + bx + c. What is vertex form good for: the pair (h, k) is the vertex of the parabola sitting in plain sight, rather than something you have to compute separately, and the shape of the equation itself tells you how the graph has been stretched and shifted from the plain y = x².
This calculator takes the standard-form coefficients a, b and c and rewrites the whole equation in vertex form quadratic notation, not just the numeric vertex point. It does this by completing the square: h = −b/2a and k = c − b²/4a, the same two values that also answer where the parabola turns.
The formula
Completing the square on ax² + bx + c factors out a, adds and subtracts (b/2a)² inside the brackets, and collects the leftover constant outside. That leaves a(x − h)² + k with h = −b/(2a) and k = c − b²/(4a). Because a is unchanged, the parabola opens upward when a is positive and downward when a is negative, exactly as it did in standard form.
| Term | Meaning |
|---|---|
| a | The leading coefficient. Controls how narrow or wide the parabola is, and which way it opens. |
| h | The x-coordinate of the vertex: h = −b/(2a). |
| k | The y-coordinate of the vertex: k = c − b²/(4a). |
| Axis of symmetry | The vertical line x = h that the parabola is a mirror image across. |
The inputs explained
| Field | What to enter |
|---|---|
| a | The coefficient of x². Cannot be zero, or the equation is a straight line, not a quadratic. |
| b | The coefficient of x. |
| c | The constant term. |
When to use it
Finding the maximum or minimum of a quadratic
Vertex form puts the turning point straight into the equation, which is often the actual question behind a word problem about maximum height, minimum cost or peak profit.
Sketching a parabola by hand
Knowing h, k and the direction it opens is usually enough to sketch a reasonable graph without plotting a table of points first.
Checking algebra homework on completing the square
Completing the square by hand is easy to get slightly wrong on the sign of h. Running the same a, b, c through here checks the working without redoing it from scratch.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the vertex form changes as a changes, with b and c fixed
The coefficient a stretched, flattened and flipped, with b and c held at 8 and 3.
| a | Vertex form | Vertex (h, k) |
|---|---|---|
| 1 | (x + 4.000)² − 13.000 | (-4.000, -13.000) |
| 2 | 2.000(x + 2.000)² − 5.000 | (-2.000, -5.000) |
| 3 | 3.000(x + 1.333)² − 2.333 | (-1.333, -2.333) |
| -1 | −(x − 4.000)² + 19.000 | (4.000, 19.000) |
| -2 | -2.000(x − 2.000)² + 11.000 | (2.000, 11.000) |
| 0.5 | 0.5000(x + 8.000)² − 29.000 | (-8.000, -29.000) |
How the vertex form changes as c changes, with a and b fixed
The constant term c shifted up and down, with a and b held at 1 and −4.
| c | Vertex form | Vertex (h, k) |
|---|---|---|
| 0 | (x − 2.000)² − 4.000 | (2.000, -4.000) |
| 2 | (x − 2.000)² − 2.000 | (2.000, -2.000) |
| 4 | (x − 2.000)² | (2.000, 0) |
| 6 | (x − 2.000)² + 2.000 | (2.000, 2.000) |
| -2 | (x − 2.000)² − 6.000 | (2.000, -6.000) |
| -4 | (x − 2.000)² − 8.000 | (2.000, -8.000) |
Questions
What is vertex form?
Vertex form writes a quadratic as a(x − h)² + k, where (h, k) is the vertex of the parabola and a controls its width and direction. It is an alternative to standard form, ax² + bx + c, that makes the turning point visible directly in the equation.
How do you convert a quadratic vertex form equation back to standard form?
Expand the squared bracket: a(x − h)² + k becomes ax² − 2ahx + ah² + k. Matching that against ax² + bx + c gives b = −2ah and c = ah² + k, which recovers the original standard-form coefficients.
Does vertex form give the same roots as standard form?
Yes, both forms describe exactly the same parabola, just written differently. The quadratic equation calculator solves for the roots and numeric vertex directly from standard form if that is what you need.
Why does completing the square work?
Adding and subtracting (b/2a)² inside the expression does not change its value, but it lets the x² and x terms be regrouped into a perfect square, a(x − h)², with everything left over collected into the constant k.
For the roots and discriminant of the same quadratic, see the quadratic equation calculator. To evaluate a general exponent or root, see the exponents and roots calculator.