# Identifying the Line of Symmetry: Definition & Examples

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: Even & Odd Functions: Definition & Examples

### You're on a roll. Keep up the good work!

Replay
Your next lesson will play in 10 seconds
• 0:02 Symmetry
• 1:39 Line of Symmetry
• 2:19 Identifying It Graphically
• 3:40 Identifying It Algebraically
• 6:53 Lesson Summary

Want to watch this again later?

Timeline
Autoplay
Autoplay

#### Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Symmetry can occur in many different ways in mathematics. In this lesson, learn how to identify unique forms of symmetry by finding the line of symmetry.

## Symmetry

You've probably seen symmetry often in real life. Every time you look in a mirror, glass, and even water, you can see a reflection of yourself. This is called symmetry. Symmetry is created when there is an exact replica or reflection of a shape or a line. You can also see symmetry in everyday objects. For example, you can take a cup and imagine a line going down the center of that cup. On each side of that line, the cup has an exact replica, or reflection, of itself.

Flowers will also often have symmetry.

There are some shapes that are symmetrical only when you draw a line in the right place. Look at this pencil.

In mathematics, we would call that horizontal line on the pencil the axis or line of symmetry. We call it that because that is the line that shows the reflection of the pencil; no other line would create a reflection on the pencil.

## What Is the Line of Symmetry?

The axis or line of symmetry is an imaginary line that runs through the center of a line or shape creating two perfectly identical halves. In higher level mathematics, you will be asked to find the axis of symmetry of a parabola.

This is a parabola, a u-shaped line on the graph.

## Identifying the Line of Symmetry Graphically

We can identify the line of symmetry graphically by simply finding the farthest point of the curve of the parabola. This is called the vertex, the point where two lines connect. If the parabola were a hill, the very highest point on that hill would represent the vertex of the parabola, or if the parabola were a valley, the very lowest spot in the valley would represent the vertex of the parabola.

Take a look at this graph. Do you see the vertex? It's at the point (2, 3).

xyxyx

Okay, now look at this graph. Can you identify the line of symmetry? The vertex is located at (-2, -4). Again, on this graph, the vertex is the very lowest point of the parabola. We can draw an imaginary line through this point to find the line of symmetry. So we would write this as y = -4.

## Identifying the Line of Symmetry Algebraically

You will most often see lines of symmetry on quadratic functions. Quadratic functions are often written in the standard form y = ax^2 + bx + c, where a, b, and c equal all real numbers. All quadratic functions create a parabola that opens up or opens down.

You can identify the line of symmetry of a quadratic function in standard form by using the formula x = -b / 2a. Let's take a look at an example.

To find the line of symmetry for this equation, you need to plug in the correct numbers into the formula. So our equation is y = 2x^2 - 4x - 3, and standard form is y = ax^2 + bx + c. So that means that a = 2 and b = -4. Let's plug that into our formula x = -b / 2a, which gives us x = -(-4/2(2)). Now evaluate the equation. 2 x 2 = 4 and -(-4)/4 = 4/4 = 1. So now we have 1, which gives us x = 1. If we were to graph this equation, it would look like this.

You can see the line of symmetry goes through the vertex (1, -5 ).

To unlock this lesson you must be a Study.com Member.

### Register for a free trial

Are you a student or a teacher?

#### See for yourself why 30 million people use Study.com

##### Become a Study.com member and start learning now.
Back
What teachers are saying about Study.com

### Earning College Credit

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.