Precalculus : Determine the Symmetry of an Equation

Study concepts, example questions & explanations for Precalculus

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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?) 

 

Possible Answers:

I don't know anything about this function. 

This isn't even a function! 

Cannot be determined from the information given 

No

Yes 

Correct answer:

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?) 

Possible Answers:

No 

There is not enough information to determine

That's not a function! 

I don't know! 

Yes 

Correct answer:

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.

Possible Answers:

Correct 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.

The correct answer is:

Example Question #21 : Graphing Functions

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

Possible Answers:

Yes

No

Not a function

Insufficient Information

Correct answer:

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.

Possible Answers:

Symmetrical about the x-axis

Symmetrical about the origin

Symmetrical about the y-axis

No symmetry

Correct answer:

Symmetrical about the x-axis

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.

 

-So there is no symmetry about the origin, and the answer is Symmetrical about the x-axis

Example Question #31 : Graphing Functions

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

Possible Answers:

There is insufficient information to determine the answer

The given information does not include a function

Yes

No

Correct answer:

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 #32 : Graphing Functions

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

Possible Answers:

No symmetry

Symmetrical about the x-axis

Symmetrical about the origin

Symmetrical about the y-axis

Correct answer:

Symmetrical about the x-axis

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.

-

-So there is no symmetry about the origin, and the answer is Symmetrical about the x-axis

Example Question #33 : Graphing Functions

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

Possible Answers:

Symmetry about the y-axis

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

Symmetry about the x-axis

Correct answer:

Symmetry about the 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, 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.

Possible Answers:

Symmetry about the y-axis and origin

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

Symmetry about the x-axis

Symmetry about the y-axis 

Symmetry about the x-axis, and y-axis

Correct answer:

Symmetry about the 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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