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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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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!

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.

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:

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.

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

Take a look at this 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.

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.

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.

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.

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.

What about identifying equations that create symmetrical lines? Take a look at this equation: *x*^4 = *y*^2 = 12. You can test to see if the equation has symmetry about the origin by replacing both the *x* and *y* values with negative *x* and *y* values.

Now, evaluate the equation. If you have a negative value with an even exponent, like *-x* to the fourth power and *-y* to the second power, you will get a positive number. Therefore, if you evaluated this equation, you would get the same equation as the original. This equation is symmetrical about the origin. We can even look at the graph of this equation and rotate it about the origin to see if the lines match up.

Okay, take a look at this equation: *x* = 2*y* - 7. Let's test this one to see if it is symmetrical about the origin. Replace the *x* and *y* values with *-x* and *-y*, evaluate the equation. We do not get the same equation; therefore, the equation is not symmetrical about the origin.

This is what the equation would look like on a graph.

Let's rotate the equation (please see the video at 05:05). You can see that the lines do not match up. So we know for sure that the equation is not symmetrical about the origin.

In conclusion, you can recognize the symmetry about the origin graphically, algebraically, and numerically. In order to identify if a graph is symmetrical, you can rotate the graph either clockwise or counterclockwise and see if the two lines or shapes match up. When you are given the points of two lines, you know that the lines are symmetrical about the origin if the points are the exact opposites of one another. To identify if an equation is symmetrical about the origin, simply replace the *x* and *y* values with a *-x* and a *-y*. Then evaluate the equation. If you get the same equation as the original, then the equation will produce a line that is symmetrical about the origin. You might want to use two tests to be sure whether or not something is symmetrical about the origin.

Following this video lesson, you should be able to:

- Describe symmetry around the origin
- Explain how to determine if a graph is symmetrical around the origin by rotating the graph or algebraically

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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