How to Use the Quadratic Vertex & Axis Calculator
The graph of a quadratic function y=ax^2+bx+c is a parabola, and its highest or lowest point is called the vertex. The vertical line that passes through the vertex and splits the parabola into two mirror-image halves is the axis of symmetry. This calculator takes coefficients a, b, and c and calculates the vertex coordinates, the axis equation, and the intervals where the function increases and decreases — all at once.
The vertex's x-coordinate comes from the formula -b/(2a); plugging that back into the original equation gives the y-coordinate. The axis equation simply reuses the vertex's x-coordinate, written as x=(vertex x-coordinate). When a is positive, the parabola opens upward, so the function decreases to the left of the vertex and increases to the right. When a is negative, it opens downward, so the pattern flips: increasing to the left and decreasing to the right.
a must be nonzero — if a is 0, the equation becomes linear (a straight line) and no vertex exists.
Frequently Asked Questions
Yes. When a is positive, the vertex's y-coordinate is the function's minimum value; when a is negative, it's the function's maximum value.
Because the axis is a vertical line. Every point on a vertical line shares the same x-coordinate, so it's written as x=(value) — meaning every point on that line has that same x value.