Determine the Symmetry of an Equation

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Pre-Calculus › Determine the Symmetry of an Equation

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

Yes

CORRECT

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 $f(-x)=(-x)^3+5(-x)-7=-x^3-5x-7\neq f(x)$

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

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

Symmetrical about the x-axis

CORRECT

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 $f(-x)=\sqrt{4+x}\neq f(x)$ so there is not symmetry about the y-axis

For a function to be symmetrical about the x-axis, it must satisfy $f(-x)=\sqrt{4+x}=-f(x)$ 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.

$(-x)=-(-y)^2+4\neq x=-y^2+4$

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

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

CORRECT

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

$f(-x)=|-x|+2\neq -f(x)$ 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.

$[(-y)=|(-x)|+2]\neq [y=|x|+2]$

So there is no symmetry about the origin.

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

$y=x^{3}+5x-7$

Yes

CORRECT

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

$f(-x)=(-x)^{3}+5(-x)-7=-x^{3}-5x-7\neq f(x)$

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

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

Yes

CORRECT

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.

Which of the following best describes the symmetry of $x=-y^{2}+4$ with respect to the x-axis, y-axis, and the origin.

Symmetrical about the x-axis

CORRECT

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

$f(-x)=\sqrt{4+x}=-f(x)$ so there is not symmetry about the y-axis

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

$f(-x)=\sqrt{4+x}=-f(x)$ 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.

$\left ( -x \right )=-(-y)^{2}+4\neq -y^{2}+4$

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

Algebraically check for symmetry with respect to the x-axis, y-axis, and the origin. $y=\left | x \right |+2$

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

CORRECT

Symmetry about the y-axis and the origin

Explanation

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

$f(x)=\left |-x \right |+2=\left |x \right |+2$ so there is symmetry about the y-axis

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

$f(-x)=\left |-x \right |+2\neq -f(x)$ 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.

$\left [ (-y)=\left |-x \right |+2 \right ]\neq \left [y=\left |x \right |+2 \right ]$

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

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

$\textup{No symmetry; swap all y variables with -y and solve for y.}$

CORRECT

$\textup{No symmetry; swap all x variables with -x and solve for x.}$

$\textup{No symmetry; replace all x and y variables with -x and -y and solve for y.}$

$\textup{Symmetrical; swap all y variables with -y and solve for y.}$

$\textup{Symmetrical; swap all x variables with -x and solve for y.}$

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.