Axis of Symmetry of a Quadratic

In the graph of a quadratic function, the axis of symmetry is an imaginary line that cuts the parabola in half to create two symmetrical sides. It goes through the vertex of the parabola so the equation of the axis of symmetry is the same as the $x$-coordinate of the vertex.

Notice that in Figure 1, the vertex is at $(-2,-2)$ and the axis of symmetry is at $x=-2$.

In Figure 2, the vertex is at $(2,7)$ and the axis of symmetry is at $x=2$.

Example

Find the vertex and axis of symmetry for $f(x)=(x+2)(x-6)$.

Step 1 -  Find the $x$-intercepts.

Set each factor equal to 0.

$x+2=0$

$x=-2$

$x-6=0$

$x=6$

Step 2 - Find the $x$-coordinate of the vertex.

The vertex is halfway between the two $x$-intercepts.

$x=6+(-2{)}^2=2$

$x=\frac{6+(-2)}{2}=2$

Step 3 - Find the $y$-coordinate of the vertex.

Substitute the $x$-coordinate into $f(x)$.

$f(2)=(2+2)(2-6)=(4)(-4)=-6$

The vertex is at $(2,-16)$.

Step 4 - Find the axis of symmetry.

The axis of symmetry is the equation where $x$ equals the $x$-coordinate of the function.

$x=2$