How this calculator works
Completing the square rewrites a quadratic so its turning point can be read directly. The coefficient a remains unchanged; h locates the axis of symmetry and k gives the minimum or maximum value. Positive a opens upward, while negative a opens downward. The sample table uses offsets of one and two from the vertex and reflects the symmetry. This tool uses ordinary floating-point numbers, so large coefficients or extreme cancellation can reduce precision.
Formula
h = -b/(2a); k = c - b^2/(4a). Conversely b = -2ah and c = ah^2 + k, giving ax^2 + bx + c = a(x - h)^2 + k.
Worked example
For y = 2x^2 - 6x + 7, h = 1.5 and k = 2.5, giving y = 2(x - 1.5)^2 + 2.5. Expanding y = -3(x + 2)^2 + 4 instead gives -3x^2 - 12x - 8.
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Frequently asked questions
Why does the sign inside the brackets look reversed?
The vertex form uses x - h. A bracket x + 2 is therefore x - (-2), so the vertex x coordinate is -2.
Can a equal zero?
No. That would make a linear or constant function rather than a parabola with a unique vertex.
Does standard form always mean ax^2 + bx + c?
Terminology varies. This page labels the expanded and vertex expressions explicitly so you can identify the intended form regardless of your textbook's naming.