Parabola Vertex and Axis of Symmetry

A parabola y = ax² + bx + c (with a not 0) always curves one direction: upward if a > 0, downward if a < 0. Because it only bends one way, it has exactly one point where it switches from falling to rising, or rising to falling. That point is the vertex. The parabola is also perfectly symmetric around a vertical line through that point - fold the graph along that line and both arms match up. That line is the axis of symmetry, and its equation is always x = (some number), since it's vertical.

Vertex form, y = a(x - h)² + k, is built to show this instantly. Since (x - h)² is never negative, the term a(x - h)² is zero exactly when x = h, making y as small as it can be (if a > 0) or as large as it can be (if a < 0) right there. So the vertex is (h, k), and the axis of symmetry is x = h. Standard form y = ax² + bx + c hides this, but you can always recover the axis with x = -b/(2a), then plug that x back into the equation to get the y-coordinate k. This connects directly to the rest of coordinate geometry: the vertex is just the point where slope momentarily equals zero, and conic sections like the ellipse and hyperbola have their own centers and axes that play the same organizing role the vertex plays here.