Axis of Symmetry
The axis of symmetry of a parabola is the vertical line that passes through the vertex, dividing the parabola into two mirror-image halves.
Formula
x = \frac{-b}{2a}
Definition
The axis of symmetry is the vertical line that cuts a parabola exactly in half, so both sides are mirror images of each other across it; for $y = ax^2 + bx + c$, it is the line $x = \frac{-b}{2a}$, passing through the vertex, and it also acts as the perpendicular bisector of any horizontal chord of the parabola. For $f(x) = a(x-h)^2 + k$ with vertex $(h,k)$, the axis $x = h$ satisfies the reflection symmetry $f(h+t) = f(h-t)$ for every $t$, since $f(h+t) = at^2 + k = f(h-t)$; this is the even part of the quadratic centered at the vertex, and it connects to the parabola's focus and directrix through the optical reflection property.
Example
For $y = x^2 - 6x + 8$, the axis of symmetry is $x = -(-6)/(2 \cdot 1) = 3$. For $y = 3x^2 - 12x + 7$, the axis is $x = 12/6 = 2$, and points $(1, -2)$ and $(3, -2)$ are equidistant from $x = 2$ with equal y-values, confirming the symmetry. For $y = 2(x - 3)^2 + 1$, axis $x = 3$: $f(5) = 2(4)+1 = 9$ and $f(1) = 2(4)+1 = 9$ match.
Key Insight
If you folded the graph along the axis of symmetry, both sides would match exactly, and the vertex always lies on it, which is why finding the vertex first and reflecting a few points halves the work of graphing. The idea generalizes to other conic sections, ellipses have two axes of symmetry and hyperbolas have two as well, connecting the algebraic structure of even functions centered at a point to geometric symmetry more broadly.