How to Test a Relation for Symmetry

When you are graphing a relation given by an equation it is helpful to find out first whether its graph has any symmetries. The most common symmetries to look for are symmetry with respect to the x and y axes and the origin.

Here are some graphs that are symmetric with respect to the y axis.

If you knew ahead of time that the graph of a relation was symmetric with respect to the y axis, it would cut your graphing work in half, because once you graphed the part where x was positive, the part to the right of the y axis, you could just reflect across the y axis to get the rest of the graph.

Here are some graphs that are symmetric with respect to the x axis.

If you knew ahead of time that the graph of a relation was symmetric with respect to the x axis, this would also cut your graphing work in half, because once you graphed the part where y was positive, the part above the x axis, you could just reflect across the x axis to get the rest of the graph.

A useful way to characterize symmetry with respect to a line is that a set of point is symmetric with respect to a line if whenever a point is in it, its reflection across the line will be in it also. The reflection of a point P across a line is the point Q such that the line is the perpendicular bisector of the segment PQ. This is just like a real world reflection that a mirror makes. If the line was a mirror, then the reflection is located at the place where the mirror image would be.

For the special lines, the y axis and the x axis there is a very easy way to find the reflection of any point. For the y axis you just change the sign of the x and for the x axis you change the sign of the y.

y axis

x axis

Here are some graphs that are symmetric with respect to the origin.

Symmetry with respect to the origin is a bit different from the others, because it is symmetry with respect to a point instead of a line, which is probably not so familiar. In this case what it comes out to mean is that if you know what the graph looks like for positive x's, then you can turn that upside down and backwards to get the graph for negative x's, so again you can cut your graphing work in half by knowing this symmetry ahead of time. The characterization of symmetry with respect to a point also has to do with reflections.

A set of points is symmetric with respect to a point if whenever a given point is in the set, its reflection across the that point is also in the set, but the idea of the reflection across a point is probably not so familiar. The reflection of a point P across a point R is the point Q such that R is the midpoint of the segment PQ.

Like with the axes, when the point that you are reflecting across is the origin there is an easy way to find the reflection. This time you change the signs of both the x and the y coordinates.

Here is a picture of all of the reflections together for quick reference.

This characterization will help us find symmetries before we graph. The idea is that for there to be a given symmetry, whenever (x,y) is in the graph, its reflection has to be also. That means that whenever (x,y) satisfies the equation, it's reflection has to satisfy it as well. So to test for a symmetry what you need to do is replace (x,y) with its reflection and see if the new equation you get is equivalent to the original. Putting this together we get the following test.

Replace (x,y) with the following.
 

y axis	(-x,y)
x axis	(x,-y)
origin	(-x,-y)

If the relation simplifies to what it was originally, then it has that symmetry.

Examples:

The instruction here is to test for symmetry with respect to the x axis, y axis, and the origin.

Example 1:

y=x²

Solution:

y axis:

y=(-x)²
y=x²

yes

Explanation:

When we replace x with -x and simplify we get the same equation as the original.

x axis:

-y=x²

no

Explanation:

When we replace y with -y we don't get the same equation as the original, and it is not equivalent to it either, because there is nothing we can do to both sides to get it to be the same. If we multiplied both sides by -1, the left side would be the same as the original, but then the right side would be different.

origin:

-y=(-x)²
-y=x²

no

Explanation:

Same as for the x axis.

Example 2:

x=y²

Solution:

y axis:

-x=y²

no

x axis:

x=(-y)²
x=y²

yes

origin:

-x=(-y)²
-x=y²

no

Example 3:

y=x³

Solution:

y axis:

y=(-x)³
y=-x³

no

x axis:

-y=x³

no

origin:

-y=(-x)³
-y=-x³
y=x³

yes

Explanation:

When we replace (x,y) with (-x,-y) we don't immediately get the same equation, but it is equivalent, because we can turn it into it by multiplying both sides by -1, so whenever (x,y) satisfies the equation (-x,-y) will also.

Example 4:

xy=5

Solution:

y axis:

(-x)y=5
-xy=5

no

x axis:

x(-y)=5
-xy=5

no

origin:

(-x)(-y)=5
xy=5

yes

Example 5:

x²+y=y³

Solution:

y axis:

(-x)²+y=y³
x²+y=y³

yes

x axis:

x²+-y=(-y)³
x²-y=-y³

no

origin:

(-x)²+(-y)=(-y)³
x²-y=-y³

no

Example 6:

y²+y=x³+x²

Solution:

y axis:

y²+y=(-x)³+(-x)²
y²+y=-x³+x²

no

x axis:

(-y)²+-y=x³+x²
y²-y=x³+x²

no

origin:

(-y)²+-y=(-x)³+(-x)²
y²-y=-x³+x²

no

Example 7:

x²+y²=1

Solution:

y axis:

(-x)²+y²=1
x²+y²=1

yes

x axis:

x²+(-y)²=1
x²+y²=1

yes

origin:

(-x)²+(-y)²=1
x²+y²=1

yes

Example 8:

y²=|x|+x

Solution:

y axis:

y²=|-x|+-x
y²=|x|-x

no

x axis:

(-y)²=|x|+x
y²=|x|+x

yes

origin:

(-y)²=|-x|+-x
y²=|x|-x

no

Example 9:

y+1=

x |x|+1

Solution:

y axis:

y+1=

-x |-x|+1

y+1=

-x |x|+1

no

x axis:

-y+1=

x |x|+1

no

origin:

-y+1=

-x |-x|+1

-y+1=

-x |x|+1

no

Explanation:

Here it almost looks like we could get there by multiplying both sides by -1, but if we did that the 1 would become minus, so it won't work.
 

Example 10:

y³=x³+1

Solution:

y axis:

y³=(-x)³+1
y³=-x³+1

no

x axis:

(-y)³=x³+1
-y³=x³+1

no

origin:

(-y)³=(-x)³+1
-y³=-x³+1

no

Explanation:

Again it won't work to multiply both sides by -1, because then the right side would be x³-1.

Even and Odd Functions

If your relation is a function, y=f(x), only symmetries with respect to the y axis and origin are interesting, because the only function that is symmetric with respect to the x axis is f(x)=0. Any other function symmetric with respect to the x axis would fail the vertical line test. For functions, symmetric with respect to the y axis and symmetric with respect to the origin have special names. They are called even and odd functions. This has something to do with the fact that for f(x)=xⁿ, f is symmetric with respect to the y axis whenever n is even, and symmetric with respect to the origin whenever n is odd.

There is a simpler test that you can use for functions. To see what it is, suppose we want to test y=f(x) for these two symmetries. To test for symmetry with respect to the y axis, we replace x with -x in the equation and see if we get an equivalent equation. This gives us

y=f(-x),

which will be equivalent to the original if and only if

f(-x)=f(x).

So a function is even if and only if

f(-x)=f(x).

For symmetry with respect to the origin we replace x with -x and y with -y and we get

-y=f(-x).

If we multiply the original equation by -1 on both sides it becomes

-y=-f(x).

These two equations will be equivalent if and only if

f(-x)=-f(x).

So a function is odd if and only if

f(-x)=-f(x).

So, to test for evenness, you plug in -x into the function formula and see if you can get it to simplify to what it was before, and to test for oddness, you plug in -x into the function formula and see if you can get it to simplify to the opposite of what it was before. One nice thing about this test is that you can test for both at once. You just replace x with -x in the function formula, and if it simplifies to what it was before, the function is even, and if you can factor out a minus and get what it was before, then it is odd. Some people like to actually write down an expression for -f(x) for comparison purposes, but this isn't really necessary.

When testing for oddness it is important to remember and be aware of the difference in meaning between the two sides of the equation. f(-x) means have you are first taking the opposite of the number and then applying to the function to it. -f(x) means first applying the function and then taking the opposite. See Using Function Notation for a review of how function notation works. What is special about an odd function is that the two will give you the same answer. To see how that works look at one of the examples that I gave at the beginning for symmetry with respect to the origin.

The values of the function at x and -x are the y values shown by the blue and red vertical lines. Notice that the  red line goes down the same amount as the blue line goes up.

For contrast we can do the same thing with an example of an even function.

Here I have drawn the lengths that correspond to f(x) and f(-x) both in blue, because they both go up, so both correspond to positive numbers, so this time f(-x) is exactly the same as f(x).

Another thing to be aware of with even and odd is not to be misled by the names. It is not like with numbers where all numbers are either even or odd. To be even or odd the function has to have a symmetry, and nowhere near all functions have symmetries. So the choice in this sort of problem is even, odd, or neither.

Examples:

The instruction for these problems is to determine whether the function is even, odd, or neither.

Example 11:

f(x)=x⁴

Solution:

f(-x)=(-x)⁴=x⁴

even

Example 12:

f(x)=x⁷

Solution:

f(-x)=(-x)⁷=-x⁷

odd

Example 13:

f(x)=x²+x³

Solution:

f(-x)=(-x)²+(-x)³=x²-x³
-f(x)=-(x²+x³)=-x²-x³

neither

Example 14:

f(x)=x|x|

Solution:

f(-x)=-x|-x|=-x|x|=-(x|x|)

odd

Example 15:

f(x)=

|x| x3+x

Solution:

f(-x)=

|-x| (-x)3+-x

=

|x| -x3-x

=

|x| -(x3+x)

=-

|x| x3+x

odd

Explanation:

Notice that with a quotient only the numerator or the denominator has to be minus for the whole thing to be minus.

Example 16:

Solution:

neither

Explanation:

The function is undefined for negative numbers, so we can't even talk about f(-x).

Example 17:

Solution:

odd

Explanation:

Odd roots are different from even roots in this way. Remember you can take an nth root of a negative number when n is odd, and the answer will be negative what you get when you take the nth root of the corresponding positive number. The cube root of -8 is -2 because -2 times -2 times -2 is -8.

Example 18:

f(x)=5

Solution:

f(-x)=5

even

Explanation:

This is a constant function, so f of anything is 5. See Using Function Notation if you need more explanation on this.