Recognizing Symmetry Graphically, Algebraically & Numerically About the Origin

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  • 0:01 Symmetry in Nature
  • 1:43 Identifying Symmetry…
  • 2:21 Identifying Symmetry…
  • 3:49 Identifying Symmetry…
  • 5:15 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.

If you look at flowers and fruits, you'll see that many of these items in nature are symmetrical. In this lesson, learn about symmetry about the origin of a graph. When you are finished, test your knowledge with a short quiz!

Symmetry in Nature

When you cut a piece of fruit in half, either side of that piece of fruit will be pretty close to identical. It will have a similar number of seeds on each side, the same lines and shape on each side, and each piece will be a reflection of the other.

In fact, sometimes a fruit will also reflect in a circle. Look at the orange below; see how it seems to be made up of several identical pieces that rotate around a center? Symmetry is very similar to cutting that piece of fruit in half. Symmetry on a graph is an exact replica or reflection of a line.

You can see symmetry when cutting fruit in half.
orange slice showing symmetry

Symmetry on a graph can appear in many different ways. It can appear on the x-axis like this, or on the y-axis like this (please see the video between 00:46 to 00:51 to see how this symmetry appears). Symmetry can also appear around the origin like this:

Symmetry around the origin
graph showing symmetry about origin

The origin is the (0, 0) point on a graph. It appears directly in the center of the graph, which is why it is called the origin.

See how the line on the graph appears to rotate around the center, just like an orange half (please see the video for this rotation at 01:07)? In order for two lines or shapes to be symmetrical about the origin, the lines or shapes must be the same distance from the center (or the origin). The two lines, or shapes, must also have the same pieces and parts mirror each other across the graph. For example, if I were to make a second line symmetrical about the origin, I would need to take this line and rotate it around.

line on a graph

Each part of the line is the mirror opposite of the line across from it.

symmetrical line on graph

Identifying Symmetry Graphically

Take a look at this graph.

squiggly lines on graph

Is it symmetrical about the origin? Let's rotate this shape around to meet the other shape (please see video at 01:51). Now that we have rotated the first shape, the two shapes match exactly.

What about this graph.

blue lines on graph

It has similar shapes, but is it symmetrical about the origin? Let's rotate the shapes (please see video at 02:08). No, the two shapes do not match up exactly after they are rotated. So, even though the two shapes on the graph are similar, they are not symmetrical about the origin.

Identifying Symmetry Numerically

You can also identify the symmetry of a line by examining the points on the two lines. If a line is symmetrical about the origin, the two lines will have points that are the exact opposite of one another.

Take this graph for example.

graph with two triangles

We have already rotated the lines of this graph and know that they are symmetrical, but let's identify some of the points on this line. Okay, now that we have the points (2, 2), (10, 2) and (6, 6), we need to know the exact opposite of the points. That would give us (-2, -2), (-10, -2) and (-6, -6). Are those points on the other side of the graph? Yes! They match perfectly on the symmetrical line.

Look at the points on this table.

points on a table

Can you identify whether these points create two symmetrical lines about the origin? Yes! The points (1, 2), (2, 4) and (3, 6) are the exact opposite from the points (-1, -2), (-2, -4), and (-3, -6). Plot those points on the graph and rotate. You can see that these points make up two lines that are symmetrical about the origin.

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