an equation is being tested for symmetry with respect to the x-axis, the y-axis, and the origin. explain why, if two of these symmetries are present then remaining one must also be present.
For x-axis reflection symmetry,
f(x,y) = f(x,-y)
For y-axis reflection symmetry,
f(x,y) = f(-x,y)
For symmetry about the origin,
f(x,y) = f(-x,-y)
If you reflect about both x and y axes, it is equivalent to reflecting about the origin.
If you reflect about the origin and then one axis, you have reflected about the other axis.
Well, think about it this way: symmetry is like having friends. If you have two friends who are super symmetrical, like a pair of identical twins, and they both go to a party, chances are they'll bring along their third twin friend too.
In the case of an equation being tested for symmetry, if it exhibits symmetry with respect to the x-axis and the y-axis, it means that when you flip it horizontally or vertically, it remains the same. These two symmetries go together like peanut butter and jelly.
Now, imagine someone tried to crash the party by saying the equation doesn't have symmetry with respect to the origin. Well, if it has symmetries with respect to both the x and y-axes, it means that when you flip it horizontally and vertically simultaneously, it should remain the same. And that, my friend, is symmetry with respect to the origin!
So, just like the third twin crashing the party, if two out of the three symmetries are present, the remaining one just naturally joins the fun. It's a symmetrical party, after all!
If an equation is being tested for symmetry with respect to the x-axis, the y-axis, and the origin, and two of these symmetries are present, it implies that the remaining one must also be present.
Let's consider each case individually:
1. Symmetry with respect to the x-axis: If a function is symmetric with respect to the x-axis, it means that if we reflect any point (x, y) on the graph across the x-axis, the new point (x, -y) would also lie on the graph. This symmetry indicates that the equation is even with respect to y, meaning it remains unchanged when we substitute y with -y.
2. Symmetry with respect to the y-axis: If a function is symmetric with respect to the y-axis, it means that if we reflect any point (x, y) on the graph across the y-axis, the new point (-x, y) would also lie on the graph. This symmetry indicates that the equation is even with respect to x, meaning it remains unchanged when we substitute x with -x.
3. Symmetry with respect to the origin: If a function is symmetric with respect to the origin, it means that if we reflect any point (x, y) on the graph across the origin, the new point (-x, -y) would also lie on the graph. This symmetry indicates that the equation is even with respect to both x and y, meaning it remains unchanged when we substitute x with -x and y with -y simultaneously.
Now, if we assume that two of these symmetries are present, it means that the equation is both even with respect to x and even with respect to y. In this case, when we substitute x with -x, the equation remains unchanged, and similarly, when we substitute y with -y, the equation also remains unchanged. Therefore, the equation must exhibit symmetry with respect to the origin as well.
Hence, if two of the symmetries (x-axis and y-axis) are present, the remaining symmetry (origin) must also be present.
If an equation is being tested for symmetry with respect to the x-axis, the y-axis, and the origin, it means we are examining whether the graph of the equation looks the same when reflected across these axes or the origin.
Let's consider the three possibilities:
1. If the equation is symmetric with respect to the x-axis, it means that for every point (x, y) on the graph, the corresponding point (x, -y) is also on the graph. In other words, the graph is unchanged when you flip it upside down. This symmetry reflects the x-axis.
2. If the equation is symmetric with respect to the y-axis, it means that for every point (x, y) on the graph, the corresponding point (-x, y) is also on the graph. In other words, the graph is unchanged when you flip it horizontally. This symmetry reflects the y-axis.
3. If the equation is symmetric with respect to the origin, it means that for every point (x, y) on the graph, the corresponding point (-x, -y) is also on the graph. In other words, the graph is unchanged when you rotate it 180 degrees around the origin. This symmetry reflects both the x-axis and y-axis simultaneously.
Now, if two of these symmetries are present, let's say there is symmetry with respect to the x-axis and the y-axis. It means that for every point (x, y) on the graph, the corresponding points (x, -y) and (-x, y) are also on the graph.
Now, if we examine the reflection across the origin (0, 0), we can see that the reflection of any point (x, y) will be (-x, -y). But according to the symmetry with respect to the x-axis and the y-axis, we know that (-x, y) and (x, -y) are on the graph. Therefore, (-x, -y) must also be on the graph.
So, if two of the symmetries are present, the third symmetry with respect to the origin must also be present. This is because the combination of the two individual symmetries ensures that their composition (reflection across the origin) is also present.
In conclusion, if an equation exhibits two symmetries out of the three possibilities mentioned, then the remaining symmetry is guaranteed to be present as well.