A quadratic function is any function of the form f(x) = ax² + bx + c, where a ≠ 0. The graph of a quadratic function is always a parabola — a U-shaped (or inverted U-shaped) curve that is symmetric about a vertical axis.
Standard Form vs Vertex Form
The standard form f(x) = ax² + bx + c is useful for finding the y-intercept (c) and the roots using the quadratic formula. The vertex form f(x) = a(x − h)² + k directly reveals the vertex (h, k) and the axis of symmetry x = h.
Finding the Vertex
The vertex is the turning point of the parabola. In standard form, the x-coordinate of the vertex is x = −b/(2a). Substitute this back into the function to find the y-coordinate. In vertex form, the vertex is simply (h, k).
Finding the Roots (Zeros)
The roots are where the parabola crosses the x-axis (f(x) = 0). Use the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. The discriminant b² − 4ac tells you how many real roots exist: positive → 2 roots, zero → 1 root (vertex touches x-axis), negative → no real roots.
How the Parameter "a" Affects the Shape
When a > 0, the parabola opens upward (minimum at vertex). When a < 0, it opens downward (maximum at vertex). The larger |a| is, the narrower the parabola. The smaller |a| is, the wider it becomes.
Try It Interactively
The best way to understand quadratic functions is to adjust the parameters yourself. In VIZMath, select the Quadratic function and use the sliders for a, b, c (or a, h, k in vertex form) to see the parabola transform in real time.