What Is the Quadratic Vertex Calculator?
Every quadratic function written in standard form, \(y = ax^2 + bx + c\), traces a parabola. The vertex is the parabola turning point — the lowest point when a is positive, or the highest point when a is negative. This calculator finds the vertex coordinates \((h, k)\) directly from the coefficients a, b and c, so you do not have to complete the square by hand.
How to Use It
Enter the three coefficients from your equation. For \(y = x^2 - 6x + 5\), set \(a = 1\), \(b = -6\) and \(c = 5\). Press calculate and the tool returns the ordered pair \((h, k)\). Remember that a must not be zero — if \(a = 0\) the expression is linear, not quadratic, and there is no parabola.
The Formula Explained
The x-coordinate of the vertex sits exactly on the axis of symmetry, given by \(h = -b / (2a)\). Plugging this value back into the equation and simplifying yields \(k = c - b^2 / (4a)\). Together these give the vertex form $$y = a(x - h)^2 + k,$$ which makes graphing, finding the maximum or minimum, and solving optimization problems straightforward.
$$\left(h,\,k\right) = \left(-\frac{b}{2\,a},\;\; c - \frac{b^{2}}{4\,a}\right)$$
Worked Example
Take \(y = 2x^2 + 8x + 3\). Here \(a = 2\), \(b = 8\), \(c = 3\). Then $$h = -8 / (2\times2) = -2,$$ and $$k = 3 - 8^2 / (4\times2) = 3 - 64/8 = 3 - 8 = -5.$$ The vertex is \((-2, -5)\), and since a is positive this is the minimum point.
FAQ
Is the vertex always a minimum? No. If \(a > 0\) the parabola opens upward and the vertex is the minimum; if \(a < 0\) it opens downward and the vertex is the maximum.
What is the axis of symmetry? It is the vertical line \(x = h\), which passes through the vertex and mirrors the two halves of the parabola.
What if a equals zero? The equation is no longer quadratic, so no vertex exists. The calculator flags this as invalid input.
