Standard To Vertex Form Calculator 

Convert any quadratic from standard form y = ax² + bx + c to vertex form y = a(x − h)² + k. This calculator shows every completing-the-square step with your numbers, plus the vertex, axis of symmetry, and intercepts.

How the conversion works

Vertex form y = a(x − h)² + k exposes the vertex (h, k) directly. The conversion is completing the square:

h = −b / 2a and k = c − b² / 4a. Equivalently, factor a from the x terms, add and subtract (b/2a)² inside the parentheses, and collect the constant terms. The calculator above shows each of these steps with your exact numbers, including exact fractions where they are clean.

The sign of a tells you the direction: a > 0 opens upward with a minimum at the vertex, a < 0 opens downward with a maximum. The axis of symmetry is the vertical line x = h, and the y-intercept is always (0, c).

Worked example

Convert y = 2x² − 8x + 6 to vertex form:

• h = −(−8) / (2 × 2) = 2; k = 6 − 64 / 8 = −2
• Complete the square: y = 2(x² − 4x) + 6 = 2[(x − 2)² − 4] + 6 = y = 2(x − 2)² − 2
• Vertex: (2, −2); axis of symmetry: x = 2; opens upward, so the minimum value is −2 at x = 2
• y-intercept: (0, 6); x-intercepts: (1, 0) and (3, 0)

These are the exact values the calculator above produces for a = 2, b = −8, c = 6.

Why vertex form is useful

Standard form is convenient for evaluating y at a given x and for applying the quadratic formula, but it hides the turning point. Vertex form shows the vertex immediately, which is exactly what optimization problems need: the maximum height of a projectile, the minimum of a cost function, or the peak of a revenue curve. Factoring into (x − h) also makes graphing by hand straightforward, since the parabola is the graph of x² scaled by a and shifted to (h, k).

Completing the square by hand

To convert manually: write y = ax² + bx + c, factor a from the first two terms to get a(x² + (b/a)x) + c, then take half of b/a, square it, and add and subtract that square inside the parentheses. The trinomial x² + (b/a)x + (b/2a)² collapses to (x + b/2a)². Multiply the subtracted square by a, combine it with c, and you have vertex form. The calculator’s step panel demonstrates this exact sequence on your coefficients.

Frequently asked questions

What is the vertex form of a quadratic?

It is y = a(x − h)² + k, where (h, k) is the vertex of the parabola. It shows the turning point and the direction of opening directly, unlike standard form y = ax² + bx + c.

How do you find h and k from standard form?

Compute h = −b / 2a, then k = c − b² / 4a. Equivalently, complete the square on ax² + bx + c. The calculator above shows both the formulas and the step-by-step algebra.

What happens if a = 0?

Then the equation is linear, not quadratic, and vertex form does not apply. The calculator rejects a = 0 and explains why.

What is the difference between vertex form and standard form?

Standard form y = ax² + bx + c is best for evaluating the function and finding the y-intercept, which is c. Vertex form y = a(x − h)² + k is best for finding the vertex, the axis of symmetry, and the maximum or minimum value.

How do I find the x-intercepts from vertex form?

Set y = 0 and solve: a(x − h)² + k = 0 gives x = h ± √(−k/a), when −k/a is non-negative. The calculator computes the intercepts from the discriminant b² − 4ac and reports zero, one, or two real intercepts.

Why do textbooks teach completing the square?

Because it is the same algebra that proves the quadratic formula, and it converts any quadratic into the form that reveals its vertex. Once you can complete the square, converting standard form to vertex form takes four short steps.