Recognizing Symmetry Graphically, Algebraically, and Numerically About the X-axis and Y-axis

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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.

When you fold a piece of paper exactly in half, you are creating symmetry. Symmetry is found in many areas of mathematics. In this lesson, learn to recognize symmetry graphically, algebraically, and numerically about the x- and y- axes.

Recognizing Symmetry

Look in a mirror. What do you see? You should see a perfect reflection of yourself and anything that is behind or around you. A reflection is an example of symmetry. The reflection you see in water, glass, or a mirror is an exact replica of you. Take a piece of paper and paint a dot on one side. While the paint is still wet, fold that paper exactly in half and then unfold it again. The dots will be a perfect reflection of each other.

In mathematics, symmetry is a perfect replica of a line or shape, only it is reversed. When looking at symmetrical lines in algebra, there are different ways that symmetry can occur. For right now, let's look at symmetry across the x- and y- axes.

This is an example of symmetry on a graph. If you fold the graph down the center line, the u-shaped line will line up perfectly with itself. It is the exact same shape on each side of the fold.

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Do you see the mirror images?

What about this one? Can you find the symmetry in this graph?

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If you fold the graph down the middle horizontal line, the u-shape will line up perfectly with itself. Just like the first graph, this graph is symmetrical because you have the same shape on each side of the fold.

What about this graph?

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This graph is not symmetrical, no matter how we fold this graph, you will not end up with two symmetrical lines.

So, why is it important to know symmetry? Well, first it helps you visualize the graphs of equations. If you are asked to graph an equation, and you already know what that equation is supposed to look like, you will know if you've graphed it right or wrong. It can also help you to plot the points from an equation faster. We'll talk more about that when we look at symmetry algebraically, but first, let's look at symmetry graphically.

Recognizing Symmetry Graphically

The x-axis is the horizontal line across the center of the graph. The x-axis helps you plot points and find information on a graph.

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This is an example of symmetry across the x-axis. Notice that the lines mirror each other right across where the x-axis lies. If this graph were on a piece of paper, then you could fold the paper right along the x-axis horizontally and have two identical looking lines on either side.

The y-axis is the line that runs vertically across the center of the graph. The y-axis helps you plot points and find information on a graph, just like the x-axis.

Symmetry across the y-axis is very similar to symmetry across the x-axis. This time, instead of looking to fold the graph along the x-axis horizontally, we are now looking to fold the graph along the y-axis vertically.

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This is an example of symmetry along the y-axis. Notice that the shape is now mirrored vertically. You can fold the graph down the center vertically, and you will have two lines that replicate or mirror one another.

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This is another example of symmetry along the y-axis. This is similar to the u-shape that you saw in the other examples. Notice that this u-shape is now facing right-side up. This is a parabola, a u-shaped line that is always symmetrical.

Recognizing Symmetry Numerically

What are some other ways you can determine if a graph is symmetrical across the x- or y-axis? Well, you can look at the points on a line to determine symmetry.

A point on a line is also known as an ordered pair and is identified by the x- and y-coordinates. Similar to map coordinates, no two coordinates are exactly the same. The x-coordinate is the first number in the ordered pair, and the y-coordinate is the second number in the ordered pair. You can use these numbers to identify symmetry.

In order to look for symmetry along the x-axis, you must multiply the y-coordinate by -1. Let's go back to a previous example. Notice the point (4, 3).

Okay, now multiply the y-coordinate, which is 3, by -1. What do you get? Right, you get -3, which gives us the point (4, -3).

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Does this point show up on our graph? Yes! It is mirrored across the x-axis perfectly. You can take any point along this line, and if you can find a matching pair across the x-axis with a negative y-coordinate, then the line is symmetrical.

Let's try to identify symmetry across the y-axis.

Take one or more points on the table. This time, we are going to multiply the x-coordinates by -1. If the line is symmetrical, you should get the remaining points.

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If we plot those points and connect them, you can see a perfect parabola that is symmetrical across the y-axis.

Recognizing Symmetry Algebraically

Okay, so what about equations? Can you tell if an equation will produce a symmetrical graph? Yes! Let's use the following equation as an example: x^3 + y^2 = 4.

In order to see if the equation is symmetrical along the x-axis, replace the positive y with a negative y.

We know that y squared is the same thing as y times y, and negative y times negative y gives us a positive y. Therefore, these equations will give us the same numbers.

If you end up with the same equation that you started with, then the equation is symmetrical across the x-axis.

Something important to note here: The equation of a line that is symmetrical across the x-axis can never be a function equation. A function is an equation that shows a relationship between the x- and the y-values. Equations that are symmetrical across the x-axis can never be functions because two y-values will always share an x-value. There is not a unique relationship between the x-values and the y-values. There can only be one y-value for each x-value.

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