# Precalculus : Symmetry

## Example Questions

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### Example Question #1 : Graphing Functions

If , what kind of symmetry does the function  have?

Even Symmetry

Symmetry across the line y=x

No Symmetry

Odd Symmetry

Even Symmetry

Explanation:

The definition of even symmetry is if

### Example Question #2 : Graphing Functions

If , what kind of symmetry does  have?

No symmetry

Even symmetry

Odd symmetry

Symmetry across the line y=x

Odd symmetry

Explanation:

is the definition of odd symmetry

### Example Question #1 : Determine The Symmetry Of An Equation

Is the following function symmetric across the y-axis? (Is it an even function?)

Yes

No

This isn't even a function!

Cannot be determined from the information given

No

Explanation:

One way to determine algebraically if a function is an even function, or symmetric about the y-axis, is to substitute  in for . When we do this, if the function is equivalent to the original, then the function is an even function. If not, it is not an even function.

For our function:

Thus the function is not symmetric about the y-axis.

### Example Question #2 : Determine The Symmetry Of An Equation

Is the following function symmetric across the y-axis? (Is it an even function?)

That's not a function!

I don't know!

No

There is not enough information to determine

Yes

Yes

Explanation:

One way to determine algebraically if a function is an even function, or symmetric about the y-axis, is to substitute  in for . When we do this, if the function is equivalent to the original, then the function is an even function. If not, it is not an even function.

For our function:

Since this matches the original, our function is symmetric across the y-axis.

### Example Question #3 : Determine The Symmetry Of An Equation

Determine if there is symmetry with the equation  to the -axis and the method used to determine the answer.

Explanation:

In order to determine if there is symmetry about the x-axis, replace all  variables with .   Solving for , if the new equation is the same as the original equation, then there is symmetry with the x-axis.

Since the original and new equations are not equivalent, there is no symmetry with the x-axis.

### Example Question #3 : Determine The Symmetry Of An Equation

Is the following function symmetrical about the y axis (is it an even function)?

Yes

Not a function

Insufficient Information

No

No

Explanation:

For a function to be even, it must satisfy the equality

Likewise if a function is even, it is symmetrical about the y-axis

Therefore, the function is not even, and so the answer is No

### Example Question #5 : Determine The Symmetry Of An Equation

Algebraically check for symmetry with respect to the x-axis, y axis, and the origin.

No symmetry

Explanation:

For a function to be symmetrical about the y-axis, it must satisfy   so there is not symmetry about the y-axis

For a function to be symmetrical about the x-axis, it must satisfy   so there is symmetry about the x-axis

For a function to be symmetrical about the origin, you must replace y with (-y) and x with (-x) and the resulting function must be equal to the original function.

### Example Question #6 : Determine The Symmetry Of An Equation

Algebraically check for symmetry with respect to the x-axis, y axis, and the origin.

Symmetry about the y-axis and origin

Symmetry about the x-axis, y-axis, and origin

Symmetry about the x-axis, and y-axis

Explanation:

For a function to be symmetrical about the y-axis, it must satisfy  so there is symmetry about the y-axis

For a function to be symmetrical about the x-axis, it must satisfy

so there is not symmetry about the x-axis

For a function to be symmetrical about the origin, you must replace y with (-y) and x with (-x) and the resulting function must be equal to the original function.

So there is no symmetry about the origin.

### Example Question #7 : Determine The Symmetry Of An Equation

Algebraically check for symmetry with respect to the x-axis, y-axis, and the origin.

Symmetry about the x-axis and y-axis

Symmetry about the y-axis and the origin

Symmetry about the x-axis, y-axis, and origin

Explanation:

For a function to be symmetrical about the y-axis, it must satisfy

so there is symmetry about the y-axis

For a function to be symmetrical about the x-axis, it must satisfy

so there is not symmetry about the x-axis

For a function to be symmetrical about the origin, you must replace y with (-y) and x with (-x) and the resulting function must be equal to the original function.

So there is no symmetry about the origin, and the credited answer is "symmetry about the y-axis".

### Example Question #8 : Determine The Symmetry Of An Equation

Which of the following best describes the symmetry of   with respect to the x-axis, y-axis, and the origin.

No symmetry

Explanation:

For a function to be symmetrical about the y-axis, it must satisfy

so there is not symmetry about the y-axis

For a function to be symmetrical about the x-axis, it must satisfy

so there is symmetry about the x-axis

For a function to be symmetrical about the origin, you must replace y with (-y) and x with (-x), and the resulting function must be equal to the original function.