Vertex Form Calculator
This vertex form calculator converts a quadratic in standard form, y equals ax squared plus bx plus c, into vertex form, y equals a times (x minus h) squared plus k, revealing the vertex and shape of the parabola in one step. You enter the three coefficients, a, b and c, and the calculator instantly returns the vertex form equation, the vertex coordinates (h, k), the turning point of the parabola, and the axis of symmetry, the vertical line x equals h that the parabola mirrors around. The method is completing the square: h works out to minus b divided by 2a, and k equals c minus b squared over 4a. Knowing the vertex shows you where the parabola turns, whether that point is a minimum or a maximum, since the sign of a decides whether it opens upward or downward, and lets you sketch or interpret the graph without plotting a table of points. This is handy for algebra homework, checking your own completing-the-square working, or reading off the maximum or minimum value of a quadratic in a physics or optimisation problem. Simply enter your three coefficients above; the only input that will not work is a equal to zero, since that would make the equation linear rather than a parabola.
The formula
For y = ax^2 + bx + c, the vertex is at h = minus b / (2a) and k = c minus b^2 / (4a). The vertex form is y = a(x minus h)^2 + k, with axis of symmetry x = h.
Worked example
For y = x^2 minus 4x + 3: h = 4 / 2 = 2 and k = 3 minus 16/4 = minus 1, so y = (x minus 2)^2 minus 1, with vertex (2, minus 1). Enter 1, -4, 3 to confirm.
Frequently asked questions
What is vertex form?
y = a(x minus h)^2 + k, which shows the vertex (h, k) directly and is got by completing the square.
How do I find the vertex?
The x coordinate is minus b over 2a; put that back into the quadratic for the y coordinate.
What does a tell me?
If a is positive the parabola opens upward with a minimum at the vertex; if negative it opens downward with a maximum.
Who this calculator is for
This calculator is for algebra students and anyone needing a quick, reliable result.
What this calculator assumes
- You enter valid numbers.
- The standard method is applied.
- Results are rounded for display.
Formula and sources
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