Symmetry

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Pre-Calculus › Symmetry

Questions 1 - 10
1

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

Yes

No

Insufficient Information

Not a function

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

2

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

Yes

No

Insufficient Information

Not a function

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

3

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

Symmetrical about the x-axis

Symmetrical about the y-axis

Symmetrical about 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.

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

4

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

Symmetrical about the x-axis

Symmetrical about the y-axis

Symmetrical about 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.

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

5

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

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

Symmetry about the x-axis

Symmetry about the x-axis, and y-axis

Symmetry about the y-axis

Symmetry about the 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.

6

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

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

Symmetry about the x-axis

Symmetry about the x-axis, and y-axis

Symmetry about the y-axis

Symmetry about the 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.

7

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

Yes

No

There is insufficient information to determine the answer

The given information does not include a function

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.

8

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

Yes

No

There is insufficient information to determine the answer

The given information does not include a function

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.

9

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

Yes

No

There is not enough information to determine

I don't know!

That's not a function!

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.

10

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

Yes

No

There is not enough information to determine

I don't know!

That's not a function!

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.

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