Axis of Symmetry
AlgebraThe 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 a vertical line that cuts the parabola exactly in half. Both sides of the parabola are mirror images of each other across this line.
Example
For $y = x^2 - 6x + 8$, the axis of symmetry is $x = -(-6)/(2 \cdot 1) = 3$. The line $x = 3$ divides the parabola perfectly in half.
Key Insight
If you folded the graph along the axis of symmetry, both sides would match exactly. The vertex always sits on the axis of symmetry.
Definition
The axis of symmetry of a parabola $y = ax^2 + bx + c$ is the vertical line $x = -b/(2a)$, which passes through the vertex. Every point on the parabola has a mirror image point equidistant from the axis on the other side. The axis of symmetry is also the perpendicular bisector of any horizontal chord of the parabola.
Example
$y = 3x^2 - 12x + 7$: axis of symmetry $x = 12/6 = 2$. Points $(1, -2)$ and $(3, -2)$ are equidistant from $x = 2$ and have equal y-values, confirming symmetry.
Key Insight
The axis of symmetry is useful for graphing: find the vertex, then plot a few points on one side and reflect them to get the other side. This halves the graphing work.
Definition
The axis of symmetry of the parabola $f(x) = a(x - h)^2 + k$ is the vertical line $x = h$, where $(h, k)$ is the vertex. Algebraically, this is the reflection symmetry: $f(h + t) = f(h - t)$ for all $t$, which follows from $f(h + t) = at^2 + k = f(h - t)$. This is the geometric manifestation of the even part of the quadratic centered at the vertex. Parabolas also have a focus and directrix related to the axis by the optical reflection property.
Example
For $y = 2(x - 3)^2 + 1$, axis is $x = 3$. Verify: $f(3 + 2) = f(5) = 2(4) + 1 = 9$ and $f(3 - 2) = f(1) = 2(4) + 1 = 9$. Equal, confirming symmetry.
Key Insight
The axis of symmetry generalizes to other conic sections: ellipses have two axes of symmetry, and hyperbolas have two as well. The axis connects algebraic structure (even functions centered at a point) to geometric symmetry.