Identifying the Line of Symmetry: Definition & Examples

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  • 0:02 Symmetry
  • 1:39 Line of Symmetry
  • 2:19 Identifying It Graphically
  • 3:40 Identifying It Algebraically
  • 6:53 Lesson Summary
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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.

A line of symmetry creates two equal sides
Sunflower with symmetry line

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

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.

Parabola

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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).

Parabola with vertex (2,3)
xyxyx

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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.

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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.

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

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