Problem 81 - Algebra and Trigonometry

Question

\(77-82\) . Test the equation for symmetry. $$ x^{4} y^{4}+x^{2} y^{2}=1 $$

Step-by-Step Solution

Answer

The equation is symmetric with respect to the x-axis, y-axis, and the origin.

Step 1: Determine Symmetry with Respect to the x-axis

To check for symmetry with respect to the x-axis, replace every instance of \(y\) with \(-y\) and check if the equation remains unchanged.\[x^{4}(-y)^{4}+x^{2}(-y)^{2}=1\]\[x^{4}y^{4}+x^{2}y^{2}=1\]The equation remains the same, thus the equation is symmetric with respect to the x-axis.

Step 2: Determine Symmetry with Respect to the y-axis

For symmetry with respect to the y-axis, replace every instance of \(x\) with \(-x\) in the equation.\[(-x)^{4}y^{4}+(-x)^{2}y^{2}=1\]\[x^{4}y^{4}+x^{2}y^{2}=1\]The equation remains unchanged, showing symmetry with respect to the y-axis.

Step 3: Determine Symmetry with Respect to the Origin

For origin symmetry, replace \(x\) with \(-x\) and \(y\) with \(-y\).\[(-x)^{4}(-y)^{4}+(-x)^{2}(-y)^{2}=1\]\[x^{4}y^{4}+x^{2}y^{2}=1\]Since substituting these values results in the original equation, it is also symmetric with respect to the origin.

Key Concepts

x-axis symmetry

To check if an equation is symmetric with respect to the x-axis, follow a simple replacement rule. Replace every instance of the variable \( y \) with \( -y \) in the given equation. If, after this substitution, the equation remains unchanged, then it is symmetric about the x-axis. This symmetry often corresponds to flipping the graph of the equation over the x-axis, like a reflection.

However, not all equations exhibit this symmetry. It’s important to carefully substitute and observe the resultant equation. In many applications, especially those involving physics and engineering, x-axis symmetry might imply certain balanced properties. For our specific case, substituting \( -y \) into the equation yields the same equation, confirming x-axis symmetry.

A key takeaway:

• Substitute \( y \) with \( -y \).
• If the original equation remains unchanged, x-axis symmetry exists.

y-axis symmetry

For y-axis symmetry, we are checking if the equation remains the same when the variable \( x \) is replaced by \( -x \). The idea is that the graph of the equation should look identical if flipped over the y-axis. This form of symmetry is quite common in functions like parabolas and other even functions.

When applying the process on our equation, we substitute \( x \) with \( -x \) and check the result. Because the powers of \( x \) are even, the substitution simplifies back to the original equation. This shows y-axis symmetry because the equation remains consistent.

Consider these steps:

• Substitute \( x \) with \( -x \).
• If the formula stays the same after this operation, y-axis symmetry is confirmed.

origin symmetry

Origin symmetry is a bit more complex, as it involves checking how the equation behaves when both variables are negated. This means replacing both \( x \) with \( -x \) and \( y \) with \( -y \). If the equation remains unchanged, the graph is symmetric about the origin, meaning you can rotate the graph 180 degrees around the origin and it will look the same.

For our current problem, when both \( x \) and \( y \) are substituted with their negatives, the equation simplifies back to its original form. This confirms symmetry about the origin.

Key steps include:

• Substitute both \( x \) with \( -x \) and \( y \) with \( -y \).
• If substitution returns the original equation, it is symmetric about the origin.