This lesson discusses how to locate the axis of symmetry of a parabola in the standard x-y coordinate plane. Learn how the vertex of the parabola relates to its axis of symmetry and how to determine the axis of symmetry from a quadratic equation.

## The Concept of Symmetry

The word symmetry implies balance. Symmetry can be applied to various contexts and situations. For instance, a marriage could be said to have symmetry if each spouse has an equal share in decision-making when it comes to money matters. But since such matters are not always clear cut, we will confine our discussion today to mathematical contexts.

Symmetry is found in geometry when a figure can be divided into two halves that are exact reflections of each other, as shown in Figure 1. These figures have line symmetry. If we were to fold each figure in half at the red lines (lines of symmetry), the two halves would lie exactly on top of each other.

## Axis of Symmetry in a Parabola

In this lesson, our concern is the **symmetry of a parabola** in the x-y coordinate plane.

Figure 2 shows a parabola that has an **axis of symmetry** that lies on the y-axis. Notice that the vertex of this parabola is at the ordered pair (0, 0). A parabola’s axis of symmetry always goes through the vertex of the parabola. In other words, it is a **vertical line** that goes through the x-coordinate of the vertex. Therefore, the equation of the axis of symmetry for this parabola is *x = 0*.

As in the geometric figures in Figure 1, if we fold the parabola at the y-axis, the two halves will lie exactly on top of each other. Parabolas always have perfect symmetry. The axis of symmetry of a parabola does not always lie on the y-axis. A parabola can have an axis of symmetry that is left or right of the y-axis, and the parabola can open upward, as in Figure 2, or it can open downward. Parabolas can also be shown as opening up to the left or right, but these types of parabolas are not considered functions and will not be a part of this lesson.

These parabolas each exhibit an axis of symmetry that does not lie on the y-axis. This parabola in Figure 3 has an axis of symmetry that intersects the x-axis at -2. Therefore, the equation of this axis of symmetry is x = -2. The parabola in Figure 4 has an axis of symmetry that intersects the x-axis at 3. Therefore, the equation of this axis of symmetry is x = 3.

A parabola is the graph of a quadratic equation. Here is the form of a **quadratic equation**:

Each of the parabolas in Figure 3 and Figure 4 can be expressed as a quadratic equation. The quadratic equation for the parabola in Figure 3 is *x^2 + 4x + 6*. The quadratic equation for the parabola in Figure 4 is *-x^2 + 6x – 8*. We can also use these quadratic equations to find the axes of symmetry of the parabolas by applying them to the equation of the axis of symmetry.

## Axis of Symmetry Equation

Here is the equation for the **axis of symmetry**:

Let’s look at the quadratic equation for the parabola in Figure 3. The a-value is 1 and the b-value is 4. The equation of the axis of symmetry is:

*x = -4/2(1) = -4/2 = -2*

This is the same value for the axis of symmetry that was exhibited by the graph. Now let’s look at the quadratic equation for the parabola in Figure 4. The a-value is -1 and the b-value is 6. Therefore, the equation of the axis of symmetry is:

*x = -6/2(-1) = -6/-2 = 3*

Again, this is the same axis of symmetry exhibited by the graph.

## The Ordered Pair of the Vertex

The axis of symmetry gives us the x-coordinate of the vertex. What about the y-coordinate of the vertex? Just plug in the value of the x-coordinate into the quadratic equation of the parabola. The parabola in Figure 3 has its vertex at x = -2. If we plug this into the quadratic equation of a parabola, we will get the y-coordinate of 2:

*y = (-2)^2 + 4(-2) + 6 = 4 – 8 + 6 = 2*

The ordered pair of the vertex is (-2, 2).

## Symmetry in a Table of Values

We can also exhibit symmetry in a table of values. Let’s use the parabola in Figure 3 again. This table shows ordered pairs for the parabola in Figure 3.

x | y |
---|---|

-5 | 11 |

-4 | 6 |

-3 | 3 |

-2 |
2 |

-1 | 3 |

0 | 6 |

1 | 11 |

The vertex is at ordered pair (-2, 2). If we move an equal number of units on either side of the axis of symmetry (x = -2), the y-coordinates will be the same. For instance, the ordered pairs (-4, 6) and (0, 6) both are two units away from the axis of symmetry and have the same y-coordinate of 6.

## Lesson Summary

Let’s review.

Characteristics of the axis of symmetry include the following:

- It is the line of symmetry of a parabola and divides a parabola into two equal halves that are reflections of each other about the line of symmetry.
- It intersects a parabola at its vertex.
- It is a vertical line with the equation of x = -b/2a.

## Learning Outcomes

Possible results of completing this lesson include the ability to:

- Determine whether something is symmetrical
- State the equation for determining the axis of symmetry
- Identify the line of symmetry of a parabola
- Calculate the vertex of a parabola