# Precalculus : Determine the Symmetry of an Equation

## Example Questions

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

Is the following function symmetric across the y-axis? (Is it an even function?) This isn't even a function!

Cannot be determined from the information given

No

Yes

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?) No

There is not enough information to determine

That's not a function!

I don't know!

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 #21 : Graphing Functions

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 #21 : Graphing Functions

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

No

Not a function

Insufficient Information

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 #22 : Graphing Functions

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 #31 : Graphing Functions

There is insufficient information to determine the answer

The given information does not include a function

Yes

No

### Example Question #32 : Graphing Functions

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 - For a function to be symmetrical about the x-axis, it must satisfy -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 #33 : Graphing Functions

Symmetry about the x-axis and y-axis

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

Symmetry about the y-axis and the origin

Explanation:

-For a function to be symmetrical about the y-axis, it must satisfy -For a function to be symmetrical about the x-axis, it must satisfy -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 #1 : 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.

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