Vertex form calculator, complete the square with steps.

Vertex form writes a quadratic as y = a(x − h)² + k, so the vertex (the turning point of the parabola) can be read straight off as (h, k). This vertex form calculator turns standard form, ax² + bx + c, into vertex form by completing the square, and shows every step with exact fractions. It also goes the other way, finds the parabola from its vertex and one point, or from its two roots, and draws the graph.

How to use the vertex form calculator

1. Pick where you start: From ax² + bx + c, Type the equation, From a, h and k, From vertex and a point, From the roots or From three points.
2. Fill in the boxes. Fractions like 5/6 and negative numbers work.
3. The big card gives the vertex form. Below it are the standard form, the vertex, the axis of symmetry, whether the parabola opens up or down, the focus and directrix, the intercepts, and the domain and range. Copy result copies them as one line, and Copy link copies a link that opens the same numbers.
4. The graph marks the vertex in orange and the x-intercepts in green. Open Steps to follow the working.

Completing the square, step by step

Take y = 2x² − 12x + 7. Take a = 2 out of the x terms: y = 2(x² − 6x) + 7. Half of −6 is −3, and (−3)² = 9. Add and take away it inside the bracket: y = 2(x² − 6x + 9) − 2 × 9 + 7. The bracket is the perfect square (x − 3)², and the leftover numbers are −18 + 7 = −11. So y = 2(x − 3)² − 11, and the vertex is (3, −11). Because a = 2 is positive, the parabola opens up and −11 is its minimum.

A shortcut with the formulas

For y = ax² + bx + c the vertex is at h = −b ÷ 2a and k = c − b² ÷ 4a, which is also what you get by putting x = h into the equation. With 2x² − 12x + 7: h = 12 ÷ 4 = 3 and k = 7 − 144 ÷ 8 = −11. The calculator uses the same numbers.

Worked example with fractions

For y = 3x² + 5x − 2: h = −5 ÷ 6 and k = −2 − 25 ÷ 12 = −49/12, so y = 3(x + 5/6)² − 49/12. The roots are x = 1/3 and x = −2, found from the same numbers, because the discriminant is 25 + 24 = 49.

Going the other way

From a = 2, h = 3 and k = −11, expand (x − 3)² = x² − 6x + 9, multiply by 2 and add k: 2x² − 12x + 18 − 11 = 2x² − 12x + 7. With a vertex (2, −3) and the point (4, 5), put the point into y = a(x − 2)² − 3: 5 = a × 4 − 3, so a = 2 and y = 2x² − 8x + 5. With roots 1 and 5 and a = −1, the parabola is y = −(x − 1)(x − 5) = −x² + 6x − 5, with its vertex halfway between the roots at (3, 4).

Typing the equation

Choose Type the equation and write the quadratic the way it appears in your book: y = 2x² − 12x + 7, f(x) = 3(x − 2)^2 + 5 or y = (x + 1)(x − 3). Brackets are multiplied out and like terms added first, so a vertex form, a factored form and a standard form all work. You can paste x² as it is, and a single other letter is read as x. Anything that is not a quadratic in x gets a plain message.

From three points

Three points with different x values fix one parabola. For (0, 3), (1, 0) and (3, 0), each point must satisfy y = ax² + bx + c. Subtracting the equations in pairs gives a = 1, b = −4 and c = 3, so y = x² − 4x + 3, which in vertex form is y = (x − 2)² − 1 with the vertex (2, −1). If two points share an x, or all three sit on one straight line, there is no such parabola and the page says which.

Focus and directrix

A parabola y = a(x − h)² + k has its focus at (h, k + 1/(4a)) and its directrix at the line y = k − 1/(4a). Every point on the curve is the same distance from the focus as from the directrix. For y = 2(x − 3)² − 11 the distance 1/(4a) is 1/8, so the focus is (3, −87/8) and the directrix is y = −89/8. The focus is on the inside of the curve, so it is above the vertex when the parabola opens up and below when it opens down.

What h and k do to the graph

Common problems

• The sign of h. In y = 2(x + 5)² + 1 the vertex is at x = −5, not 5. The form has x − h, so x + 5 means h = −5.
• Forgetting to multiply by a. What you add inside the bracket is multiplied by a when it comes out. The steps show that line.
• A vertex point on its own. A vertex and a second point at the same x fix nothing, and the calculator asks for another point.

To find where the parabola crosses the x axis, use the quadratic formula calculator. To plot it with other curves, use the graphing calculator.