Question

Algebraically determine if the relation x2y2-x4=1 is symmetrical with respect to the x-axis, y-axis, or the origin.

Answer 1

Hope this helped!

For symmetry with respect to the Y-Axis, check to see if the equation is the same when we replace x with −x:

Example: is y = x2 symmetric about the y-axis?

Try to replace x with −x:

y = (−x)2

Since (−x)2 = x2 (multiplying a negative times a neagtive gives a positive), there is no change

So y = x2 is symmetric about the y-axis

For Symmetry About X-Axis

Use the same idea as for the Y-Axis, but try replacing y with −y.

Example: is y = x3 symmetric about the x-axis?

Try to replace y with −y: −y = x3

Now try to get the original equation:

Try multiplying both sides by −1: y = −x3

It is different.

So y = x3 is not symmetric about the y-axis

Diagonal Symmetry

Try swapping y and x (i.e. replace both y with x and x with y).

Example: does y = 1/x have Diagonal Symmetry?

Start with: y = 1/x

Try swapping y with x: x = 1/y

Now rearrange that: multiply both sides by y: xy = 1

Then divide both sides by x: y = 1/x

And we have the original equation. They are the same.

So y = 1/x has Diagonal Symmetry

Origin Symmetry

origin symmetry

Origin Symmetry is when every part has a matching part:

the same distance from the central point

but in the opposite direction.

Check to see if the equation is the same when we replace both x with −x and y with −y.

Example: does y = 1/x have Origin Symmetry?

Start with:

y = 1/x

Replace x with −x and y with −y:

(−y) = 1/(−x)

Multiply both sides by −1:

y = 1/x

And we have the original equation.

So y = 1/x has Origin Symmetry

Amazing! y = 1/x has origin symmetry as well as diagonal symmetry!

Answer 2

Answer:

Step-by-step explanation:

D. origin symmetry