Problem 82 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph is symmetric about the origin.

Step 1: Test for Symmetry about the x-axis

To test for symmetry about the x-axis, replace \( y \) with \( -y \) in the equation and simplify.Original Equation: \[ x^2y^2 + xy = 1 \]Substitute \( y = -y \):\[ x^2(-y)^2 + x(-y) = 1 \]\[ x^2y^2 - xy = 1 \]The new equation is different from the original, so the graph is not symmetric about the x-axis.

Step 2: Test for Symmetry about the y-axis

To test for symmetry about the y-axis, replace \( x \) with \( -x \) in the equation and simplify.Original Equation: \[ x^2y^2 + xy = 1 \]Substitute \( x = -x \):\[ (-x)^2y^2 + (-x)y = 1 \]\[ x^2y^2 - xy = 1 \]The new equation is different from the original, so the graph is not symmetric about the y-axis.

Step 3: Test for Symmetry about the Origin

To test for symmetry about the origin, replace both \( x \) with \( -x \) and \( y \) with \( -y \) in the equation and simplify.Original Equation: \[ x^2y^2 + xy = 1 \]Substitute \( x = -x \) and \( y = -y \):\[ (-x)^2(-y)^2 + (-x)(-y) = 1 \]\[ x^2y^2 + xy = 1 \]The new equation is the same as the original, so the graph is symmetric about the origin.

Key Concepts

x-axis symmetry

When talking about x-axis symmetry in equations, we want to check if the equation remains unchanged when we reflect it over the x-axis. This means the top half of the graph looks the same as the bottom half. To test this, we replace every instance of the variable \( y \) with \( -y \) in the equation.

For example, in the equation \( x^2 y^2 + xy = 1 \), we substitute \( y \) with \( -y \):

• The substitution gives us: \( x^2(-y)^2 + x(-y) = 1 \)
• Since \((-y)^2 = y^2\), it simplifies to: \( x^2 y^2 - xy = 1 \)

Comparing this result to the original equation, \( x^2 y^2 + xy = 1 \), we see they are not the same (the signs are different). This means the equation does not have x-axis symmetry. It's like trying to check if a butterfly's reflection in water would look the same—since it doesn't in this case, there's no symmetry.

y-axis symmetry

Y-axis symmetry is another way to explore symmetry in equations. It involves reflecting a graph over the y-axis and checking if the reflected graph looks identical to the original. To test this, we replace the variable \( x \) with \( -x \) in our equation.

For the equation \( x^2 y^2 + xy = 1 \), we do the following:

• Substitute \( x \) with \( -x \): \( (-x)^2 y^2 + (-x)y = 1 \)
• \((-x)^2\) equals \( x^2 \), so the equation becomes: \( x^2 y^2 - xy = 1 \)

Compare this with the original equation \( x^2 y^2 + xy = 1 \). Here, again, the equations are not identical because the sign of the last term has changed. Thus, the graph of this equation does not have y-axis symmetry. Consider it like flipping a pancake – if the two sides don't match, there's no symmetry here either.

origin symmetry

Origin symmetry in equations means we reflect the graph over both the x-axis and y-axis simultaneously, essentially rotating it 180 degrees around the origin. For a graph to be symmetric about the origin, its equation must remain unchanged after substituting both variables \( x \) and \( y \) with their opposites \(-x\) and \(-y\).

In our equation \( x^2 y^2 + xy = 1 \), this process is:

• Substitute \( x \) with \( -x \) and \( y \) with \( -y \): \( (-x)^2(-y)^2 + (-x)(-y) = 1 \)
• Since \((-x)^2 = x^2\) and \((-y)^2 = y^2\), the equation simplifies back to: \( x^2 y^2 + xy = 1 \)

The new equation is exactly the same as the original, signaling that the graph is symmetrically centered at the origin. Think of it like a point in the center where you can flip the graph in any direction and it will look the same; that's the hallmark of origin symmetry.