Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Check for symmetry with respect to both axes and the origin. \(x y=2\)

Step-by-Step Solution

Answer

The given equation \(x y=2\) is not symmetric with respect to either the x-axis or the y-axis,but it is symmetric with respect to the origin.

Step 1: Check for Symmetry about the x-axis

Replace \(y\) with \(-y\) in the equation and see if the resulting equation is equivalent to the original equation. If \(x (-y) = 2\), then this simplifies to \(-x y = 2\), which is not equivalent to the original equation \(x y = 2\). Therefore, the equation is not symmetric about the x-axis.

Step 2: Check for Symmetry about the y-axis

Replace \(x\) with \(-x\) in the equation and see if the resulting equation is equivalent to the original equation. If \((-x) y = 2\), then this simplifies to \(-x y = 2\), which is not equivalent to the original equation \(x y = 2\). Therefore, the equation is not symmetric about the y-axis.

Step 3: Check for Symmetry about the Origin

Replace both \(x\) and \(y\) with \(-x\) and \(-y\), respectively, in the equation and see if the resulting equation is equivalent to the original equation. If \((-x)(-y) = 2\), then this simplifies to \(x y = 2\), which is equivalent to the original equation. Therefore, the equation is symmetric about the origin.

Key Concepts

Understanding X-Axis Symmetry

To determine if a graph displays x-axis symmetry, replace the variable \( y \) with \( -y \) in the equation. This transformation checks whether flipping the graph over the x-axis results in the same graph.

• If the resulting equation matches the original, the graph is symmetric with respect to the x-axis.
• If it does not, as in the case of \( xy = 2 \) which becomes \(-xy = 2\), then the graph does not possess x-axis symmetry.

This kind of symmetry means that for every point \( (x, y) \), there is a corresponding point \( (x, -y) \) on the graph.

Exploring Y-Axis Symmetry

Y-axis symmetry involves replacing the \( x \) variable with \( -x \) in your equation. By doing this, we are testing if reflecting the graph across the y-axis retains the original equation.

• If the transformed equation equals the original, then y-axis symmetry exists.
• For the equation \( xy = 2 \), replacing \( x \) with \( -x \) results in \(-xy = 2\), which is not the same as the original equation.

Thus, the graph of \( xy = 2 \) is not symmetric about the y-axis. Y-axis symmetry implies that for each point \( (x, y) \), there is an equivalent point \( (-x, y) \) on the graph.

Demystifying Origin Symmetry

Origin symmetry can be a bit trickier to spot than other types. This involves replacing both \( x \) with \( -x \) and \( y \) with \( -y \). Essentially, we are checking if a 180-degree rotation about the origin brings us back to our original graph.

• Should the modified equation be equivalent to the original, then the equation has origin symmetry.
• For example, transforming \( xy = 2 \) to \((-x)(-y) = 2\) simplifies back to \(xy = 2\).

This confirms that the graph for \( xy = 2 \) is symmetric about the origin. Origin symmetry suggests that for every point \( (x, y)\), there is a corresponding point \( (-x, -y) \). This type of symmetry indicates that the graph can be effectively rotated around the origin without changing its appearance.