91Ó°ÊÓ

In mathematical terms, a graph is said to exhibit x-axis symmetry if every point

• (x, y)

on the graph has a corresponding point

• (x, -y)

on the graph. This means that if you flip the graph over the x-axis, it will look exactly the same. To test for x-axis symmetry, you substitute every instance of \( y \) with \( -y \) in the equation and check if the equation remains unchanged. If it does, the graph is symmetric with respect to the x-axis.
For example, let's consider the given equation: \( x^{2}y^{2} + xy = 1 \). By substituting \( y \)with \( -y \), we get \( x^{2}(-y)^{2} + x(-y) = 1 \), which simplifies to \( x^{2}y^{2} - xy = 1 \).
Since this new equation is different from the original, we can conclude that the given equation does not have x-axis symmetry. This test tells us that reflecting the graph over the x-axis does not produce the same set of points.

Y-axis symmetry occurs when a graph is mirrored perfectly over the y-axis. This means that for every point

• (x, y)

on the graph, there is a corresponding point

• (-x, y)

also present.
To verify y-axis symmetry, one substitutes \( x \) with \( -x \) in the original equation and checks if this change yields an identical equation. If the equation remains unchanged, it confirms y-axis symmetry.
Take the equation: \( x^{2}y^{2} + xy = 1 \). By making the substitution \( x \) with \( -x \), the equation becomes \( (-x)^{2}y^{2} + (-x)y = 1 \), which simplifies to \( x^{2}y^{2} - xy = 1 \).
With the equation differing from the original, we determine that the given equation does not exhibit y-axis symmetry. Understanding this concept helps us realize that reflection around the y-axis will not produce the same graph.

A graph exhibits origin symmetry if the graph appears unchanged when rotated 180° around the origin. For such symmetry, each point on the graph

• (x, y)

can be transformed to

• (-x, -y)

and still belong on the graph.
To test for origin symmetry, substitute both \( x \) with \( -x \) and \( y \) with \( -y \) in the equation simultaneously. If the equation remains identical after manipulation, it is symmetric about the origin.
Consider the equation: \( x^{2}y^{2} + xy = 1 \). Upon substituting both variables, \( (-x)^{2}(-y)^{2} + (-x)(-y) = 1 \), it simplifies to \( x^{2}y^{2} + xy = 1 \).
Here, the equation remains unchanged, confirming that our given equation \( x^{2}y^{2} + xy = 1 \) possesses origin symmetry. This insight tells us that the graph can be flipped around the origin without altering its appearance.