Graphing Parabolas

The graph of a quadratic is a parabola, a U-shape.

• If a > 0 it opens up and the vertex is the minimum.
• If a < 0 it opens down and the vertex is the maximum.
• Larger |a| makes it narrower.

Find the vertex first. Its x-coordinate is −b/(2a); substitute back to get the y-coordinate.

The axis of symmetry is the vertical line through the vertex, x = −b/(2a). The parabola mirrors across it, so once you plot points on one side you get the other side free.

The y-intercept is c, available with no work at all.

A quadratic function f(x) = ax² + bx + c with a ≠ 0 graphs as a parabola with axis of symmetry x = −b/(2a) and vertex (−b/(2a), f(−b/(2a))). The axis formula follows from completing the square, which rewrites the function in vertex form a(x − h)² + k with h = −b/(2a).

The parameter a controls both direction and vertical stretch; it is the same a as in the transformation a·f(x) applied to the parent function y = x².

The range follows from the vertex: [k, ∞) for a > 0 and (−∞, k] for a < 0. The domain is always ℝ.

Zeros, if real, are symmetric about the axis, so their average is exactly −b/(2a) — which provides both a fast route to the vertex once you have the roots and a check on the roots once you have the vertex. The number of real zeros is governed by the discriminant b² − 4ac.