Vertex Form Calculator

Convert a quadratic from standard form y = ax² + bx + c to vertex form y = a(x − h)² + k, or the other way round. The calculator shows the completing-the-square steps, the vertex, axis of symmetry, intercepts, domain and range. Example: y = x² − 6x + 5 becomes y = (x − 3)² − 4, with vertex (3, −4) and roots x = 1 and x = 5.

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Convert

b (standard form)

c (standard form)

h (vertex form)

k (vertex form)

How to use the Vertex Form Calculator

Choose the direction: standard to vertex form, or vertex to standard form.
Enter a, b and c (standard) or a, h and k (vertex).
Press Calculate to see both forms, the vertex, axis, intercepts, domain and range.
Open Show steps to follow the completing-the-square work.

Formula and method

Vertex form: y = a(x − h)² + k, where the vertex is (h, k).

From standard form y = ax² + bx + c: h = −b / 2a and k = c − b² / 4a (equivalently k = f(h)).

Example: y = x² − 6x + 5. h = 6 / 2 = 3, k = 5 − 36 / 4 = −4. Vertex form: y = (x − 3)² − 4.

From vertex form back to standard: b = −2ah and c = ah² + k.

What is vertex form?

Vertex form shows the turning point of a parabola directly. The number h shifts the graph right (positive h) or left, k shifts it up or down, and a stretches it and sets the direction. Standard form is better for finding the y-intercept; vertex form is better for graphing and for maximum or minimum problems.

Completing the square, step by step

1) Factor a out of the x² and x terms. 2) Take half of the coefficient of x inside the brackets and square it. 3) Add and subtract that amount, keeping the equation balanced by multiplying the subtracted piece by a. 4) Rewrite the first three terms as a perfect square. The calculator prints every step with your numbers.

Worked example with a not equal to 1

y = 2x² + 8x + 3. Factor: 2(x² + 4x) + 3. Half of 4 is 2, and 2² = 4: 2(x² + 4x + 4) + 3 − 8 = 2(x + 2)² − 5. The vertex is (−2, −5), the axis is x = −2, and the parabola opens upward.

Reading the vertex

In y = a(x − h)² + k the h has the opposite sign inside the bracket: (x + 2) means h = −2. This sign flip is the most common mistake.

Vertex form	Vertex	Axis
y = (x − 3)² − 4	(3, −4)	x = 3
y = −2(x + 1)² + 5	(−1, 5)	x = −1
y = ½(x − 4)²	(4, 0)	x = 4
y = 3(x + 2)² − 1	(−2, −1)	x = −2

Maximum and minimum

If a is positive the vertex is the lowest point (minimum value k). If a is negative the vertex is the highest point (maximum value k). Word problems about projectile height or revenue use this directly.

Finding roots from vertex form

Set y = 0 and solve: (x − h)² = −k / a. There are real roots only if −k / a is zero or positive. The roots are x = h ± √(−k / a).

Standard form vs vertex form vs factored form

Factored form a(x − r₁)(x − r₂) shows the roots. Standard form shows the y-intercept. Vertex form shows the turning point. All three describe the same graph.

Accuracy and limits

Results are exact arithmetic shown to four decimals. If your answer has fractions, check the decimal against h = −b / 2a.

Frequently asked questions

How do I convert to vertex form?

Compute h = −b/(2a) and k = c − b²/(4a), then write y = a(x − h)² + k.

What is the vertex of y = x² − 6x + 5?

(3, −4).

Why is h written with the opposite sign?

Because (x − h) is zero when x = h. A vertex at x = −2 is written (x + 2).

What does a tell me?

Its sign sets the direction (up or down) and its size sets the width.

Can I get back to standard form?

Yes. Choose Vertex → standard, then enter a, h and k.

Sources

Quadratic function (Wikipedia)

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Effortless Math Team

3 hours ago