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Example 1 Determine
the symmetry of each of the following equations.
(a)  [Solution]
(b) [Solution]
(c) [Solution]
(d) [Solution]
(e) [Solution]
Solution
(a)
We’ll first check for symmetry about the x-axis. This means that we need to replace all the y’s with y.
That’s easy enough to do in this case since there is only one y.
Now, this is not an equivalent equation since the terms on
the right are identical to the original equation and the term on the left is
the opposite sign. So, this equation
doesn’t have symmetry about the x-axis.
Next, let’s check symmetry about the y-axis. Here we’ll replace
all x’s with x.
After simplifying we got exactly the same equation back
out which means that the two are equivalent.
Therefore, this equation does have symmetry about the y-axis.
Finally, we need to check for symmetry about the
origin. Here we replace both
variables.
So, as with the first test, the left side is different
from the original equation and the right side is identical to the original
equation. Therefore, this isn’t
equivalent to the original equation and we don’t have symmetry about the
origin.
[Return to Problems]
(b)
We’ll not put in quite as much detail here. First, we’ll check for symmetry about the x-axis.
We don’t have symmetry here since the one side is
identical to the original equation and the other isn’t. So, we don’t have
symmetry about the x-axis.
Next, check for symmetry about the y-axis.
Remember that if we take a negative to an odd power the
minus sign can come out in front. So,
upon simplifying we get the left side to be identical to the original
equation, but the right side is now the opposite sign from the original
equation and so this isn’t equivalent to the original equation and so we
don’t have symmetry about the y-axis.
Finally, let’s check symmetry about the origin.
Now, this time notice that all the signs in this equation
are exactly the opposite form the original equation. This means that it IS equivalent to the
original equation since all we would need to do is multiply the whole thing
by “-1” to get back to the original equation.
Therefore, in this case we have symmetry about the origin.
[Return to Problems]
(c)
First, check for symmetry about the x-axis.
This is identical to the original equation and so we have
symmetry about the x-axis.
Now, check for symmetry about the y-axis.
So, some terms have the same sign as the original equation
and other don’t so there isn’t symmetry about the y-axis.
Finally, check for symmetry about the origin.
Again, this is not the same as the original equation and
isn’t exactly the opposite sign from the original equation and so isn’t
symmetric about the origin.
[Return to Problems]
(d)
First, symmetry about the x-axis.
It looks like no symmetry about the x-axis
Next, symmetry about the y-axis.
So, no symmetry here either.
Finally, symmetry about the origin.
And again, no symmetry here either.
This function has no symmetry of any kind. That’s not unusual as most functions don’t
have any of these symmetries.
[Return to Problems]
(e)
Check x-axis
symmetry first.
So, it’s got symmetry about the x-axis symmetry.
Next, check for y-axis
symmetry.
Looks like it’s also got y-axis symmetry.
Finally, symmetry about the origin.
So, it’s also got symmetry about the origin.
Note that this is a circle centered at the origin and as
noted when we first started talking about symmetry it does have all three
symmetries.
[Return to Problems]
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