## Finding the Axis of Symmetry and Vertex of a Parabola

Look again at the figure below. Do you see that we could fold each parabola in half and that one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the **axis of symmetry** of the parabola.

We show the same two graphs again with the axis of symmetry in red. See the figure below.

The equation of the **axis of symmetry** can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of \(y=a{x}^{2}+bx+c\) is \(x=-\frac{b}{2a}.\)

So, to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula \(x=-\frac{b}{2a}\).

Look back at the first figure above. Are these the equations of the dashed red lines?

The point on the parabola that is on the axis of symmetry is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards. This point is called the **vertex** of the parabola.

We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means its *x*-coordinate is \(-\frac{b}{2a}\). To find the *y*-coordinate of the vertex, we substitute the value of the *x*-coordinate into the quadratic equation.

### Axis of Symmetry and Vertex of a Parabola

For a parabola with equation \(y=a{x}^{2}+bx+c\):

- The axis of symmetry of a parabola is the line \(x=-\frac{b}{2a}\).
- The vertex is on the axis of symmetry, so its
*x*-coordinate is \(-\frac{b}{2a}\).

To find the *y*-coordinate of the vertex, we substitute \(x=-\frac{b}{2a}\) into the quadratic equation.

## Example

For the parabola \(y=3{x}^{2}-6x+2\) find: the axis of symmetry and the vertex.

### Solution

The axis of symmetry is the line \(x=-\frac{b}{2a}\). | ||

Substitute the values of a, b into the equation. | ||

Simplify. | \(x=1\) | |

The axis of symmetry is the line \(x=1\). | ||

The vertex is on the line of symmetry, so its x-coordinate will be \(x=1\). | ||

Substitute\(x=1\) into the equation and solve for y. | ||

Simplify. | ||

This is the y-coordinate. | \(y=-1\) The vertex is \(\left(1,\text{−}1\right).\) |