Question

An equation is being tested for symmetry with respect to the \(x\) -axis, the \(y\) -axis, and the origin. Explain why, if two of these symmetries are present, the remaining one must also be present.

Short answer

If two of the symmetries (x-axis, y-axis, origin) are present, the third must be present because the combination of two transformations inherently includes the third.

Step-by-step solution

- Understand the Symmetries

For a function to be symmetric with respect to the x-axis, replacing y with -y should yield the same equation. For y-axis symmetry, replacing x with -x should yield the same equation. For symmetry with respect to the origin, replacing both x with -x and y with -y should yield the same equation.

- Symmetry with respect to the x-axis and y-axis

Assume the function is symmetric with respect to the x-axis and the y-axis. This means that the transformations y to -y and x to -x do not change the equation.

- Combine Transformations

Combine the two transformations: first apply y to -y (x-axis symmetry), then apply x to -x (y-axis symmetry). The combination of these transformations results in changing (x, y) to (-x, -y), which is the criterion for symmetry with respect to the origin.

- Symmetry with respect to x-axis and origin

Consider the function is symmetric with respect to the x-axis and the origin. This means replacing y with -y and both x and y with -x and -y do not change the equation. From the transformation for origin symmetry, we can see it effectively includes transformations for both the x and y axes.

- Symmetry with respect to y-axis and origin

Assume the function is symmetric with respect to the y-axis and the origin. Given the symmetry to y-axis means replacing x with -x doesn't change the equation. Combining this with origin symmetry implies replacing both x and y with -x and -y, which naturally indicates symmetry with respect to the x-axis.

- Conclude

Based on the steps, if any two symmetries (x-axis, y-axis, origin) are present, transforming the variables consistently demonstrates that the third symmetry must also exist.

Key concepts

Understanding Symmetries in Equations

When analyzing the graphical representation of equations, understanding symmetries can provide significant insights. There are three main types of symmetries to consider: x-axis symmetry, y-axis symmetry, and origin symmetry. These types of symmetry simplify understanding the shapes and behaviors of functions without manually plotting numerous points.

x-axis symmetry

An equation exhibits x-axis symmetry if reflecting the graph over the x-axis leaves the graph unchanged. This means that for every point \( (x, y) \) on the graph, the point \( (x, -y) \) is also on the graph.
In mathematical terms, this symmetry is verified by replacing \ y \ with \ -y \ and checking if the equation remains unchanged. For example, if we have a function \( y = f(x) \), to check for x-axis symmetry, we test whether \( -y = f(x) \) also holds true.

y-axis symmetry

An equation has y-axis symmetry if reflecting the graph over the y-axis leaves the graph unchanged. This implies that for every point \( (x, y) \) on the graph, the point \( (-x, y) \) is also on the graph.
To determine y-axis symmetry mathematically, replace \ x \ with \ -x \ in the equation and check if the equation remains the same. For example, if our equation is \( y = f(x) \), we need to verify if \( y = f(-x) \) is also true.

origin symmetry

A function has origin symmetry if rotating the graph 180 degrees about the origin leaves the graph unchanged. This criterion means for every point \( (x, y) \) on the graph, the point \( (-x, -y) \) is also on the graph.
To check for origin symmetry, replace both \ x \ with \ -x \ and \ y \ with \ -y \ in the equation. For instance, if the equation is \( y = f(x) \), we need to see whether \( -y = f(-x) \) holds true.