What This Calculator Does
This tool finds the vertex of a parabola given a quadratic in standard form, f(x) = ax² + bx + c. The vertex is the turning point of the parabola — its lowest point when the curve opens upward, or its highest point when it opens downward. The vertex is written as the ordered pair (h, k), where h is the x-coordinate and k is the y-coordinate.
How to Use It
Enter the three coefficients a, b, and c from your equation. The coefficient a must not be zero (otherwise the equation is linear, not quadratic). The calculator returns the vertex (h, k), the axis of symmetry (x = h), and whether the parabola opens upward or downward.
The Formula Explained
The x-coordinate of the vertex comes from \(h = -\frac{b}{2a}\), the same value as the axis of symmetry. Substituting h back into the function gives the y-coordinate, which simplifies to \(k = c - \frac{b^{2}}{4a}\). When a > 0 the parabola opens upward and k is a minimum; when a < 0 it opens downward and k is a maximum.
$$\left(h,\,k\right) = \left(-\frac{\text{b}}{2\,\text{a}},\;\; \text{c} - \frac{\text{b}^{2}}{4\,\text{a}}\right)$$
Worked Example
Take f(x) = x² − 4x + 3, so a = 1, b = −4, c = 3. Then $$h = -\frac{-4}{2\cdot 1} = \frac{4}{2} = 2.$$ And $$k = 3 - \frac{(-4)^{2}}{4\cdot 1} = 3 - \frac{16}{4} = 3 - 4 = -1.$$ The vertex is (2, −1), the axis of symmetry is x = 2, and since a is positive the parabola opens upward with a minimum of −1.
FAQ
What is vertex form? The vertex form is f(x) = a(x − h)² + k. Once you know (h, k) from this calculator, you can rewrite the standard form directly into vertex form using the same a.
Why must a not be zero? If a = 0 there is no x² term, so the graph is a straight line and has no vertex.
Is the axis of symmetry the same as h? Yes. The axis of symmetry is the vertical line x = h that splits the parabola into two mirror-image halves.
