Back To CourseMath 105: Precalculus Algebra
14 chapters | 124 lessons | 12 flashcard sets
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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.
Do you see the mirror images?
What about this one? Can you find the symmetry in this graph?
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?
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.
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.
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.
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.
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.
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).
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.
If we plot those points and connect them, you can see a perfect parabola that is symmetrical across the y-axis.
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.
We can use the vertical line test to see if a line on a graph is a function. This line is not a function because the vertical line hits more than one point.
In fact, no matter where we put a vertical line, the line will always hit more than one point. If any vertical line hits more than one point, you do not have a function equation.
Why is this? It has to do with the relationship between two numbers. Let's say you are watering your yard a lot, or filling up a pool. You know that when you use a lot of water, your water bill will go up. The amount of water you use has a unique relationship with the amount your water bill will cost. That is an example of a function. You can't use the same amount of water and have a bill with two different prices - how would you know which one to pay?
You can find symmetry across the y-axis in equations by replacing the x with a negative x and evaluating the equation. Try finding y-axis symmetry in this equation: y = x^2 + 7.
First, replace the x with a negative x, and then evaluate the equation. In this equation, x is squared. Therefore, negative x times negative x equals a positive x. Even though we replaced x with a negative x, we still get the same equation. This equation is symmetrical across the y-axis.
You can test an equation to see if it is symmetrical across either the x- or the y-axis. Try this equation: y^2 - x^2 = 16. First, test to see if it is symmetrical across the x-axis by replacing positive y with a negative y and evaluating the equation.
We get the same equation, so this equation is symmetrical across the x-axis. Now, test to see if the equation is symmetrical across the y-axis by replacing positive x with a negative x and evaluating the equation.
We also get the same equation; this means that the equation is also symmetrical across the y-axis. But how can this be?
Look at this graph. The graph shows a line that can be folded both horizontally and vertically with symmetrical lines on each side of the x- and y-axis. Therefore, the graph is symmetrical for both the x-axis and the y-axis.
Let's review. Symmetry is when there is a perfect replica of a line or a shape. You can visually identify symmetry graphically by folding the graph along either the x- or the y-axis.
The x-axis is the horizontal line that goes across the center of the graph, so if you have a line that is symmetric on either side of this line, then it is symmetric across the x-axis.
The same goes for the y-axis, which is the vertical line that goes across the center of the graph. If you have a line that is symmetric on either side of the y-axis, then it is considered symmetric across the y-axis.
You can also identify symmetry numerically by examining the points of a graph. If each point has a mirror opposite, negative and positive y- or x-coordinates, then there is symmetry. The y-coordinate is the second number that appears in a point, or ordered pair.
Lines that are symmetrical along the x-axis will have y-coordinates that are the same positive and negative numbers. Lines that are symmetrical along the y-axis will have x-coordinates that are the same positive and negative numbers.
You can test equations for symmetry by replacing either the y- or x-values in the equation with negative y- or x-values. If you replace positive y with negative y and get the original equation, then you have symmetry across the x-axis. If you replace positive x with negative x and get the same equation, then you have symmetry across the y-axis.
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Back To CourseMath 105: Precalculus Algebra
14 chapters | 124 lessons | 12 flashcard sets