Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Check for symmetry with respect to both axes and the origin. \(x^{3} y=1\)

Step-by-Step Solution

Answer

The function \(x^{3} y = 1\) is symmetric about the origin, but not symmetric with respect to the x-axis or the y-axis.

Step 1: Symmetry about the X-axis

For symmetry about the x-axis, substitute \(y\) with \(-y\) in the function: \((-x)^{3} (-y) = 1\). This does not simplify to \(x^{3}y = 1\), indicating the function lacks symmetry about the x-axis.

Step 2: Symmetry about the Y-axis

For symmetry about the y-axis, substitute \(x\) with \(-x\): \[(-x)^{3} y = -x^{3}y = 1\]. This does not simplify to \(x^{3} y = 1\), so the function also lacks symmetry about the y-axis.

Step 3: Symmetry about the Origin

For symmetry about the origin, substitute \(x\) with \(-x\) and \(y\) with \(-y\): \[(-x)^{3}(-y) = x^{3}y = 1\]. This simplifies to the original function, \(x^{3} y = 1\), so the function is symmetric about the origin.

Key Concepts

x-axis symmetry

To determine if a function has x-axis symmetry, we replace every occurrence of \(y\) in the equation with \(-y\). After making this substitution, if the resulting equation is equivalent to the original equation, then the function is symmetric with respect to the x-axis.

For example, consider the function \(x^{3}y = 1\). We substitute \(-y\) for \(y\) to get \((-x)^{3}(-y) = 1\). Simplifying this, we find that it does not reduce back to the original equation \(x^{3}y = 1\). Hence, the function does not have x-axis symmetry.

Understanding x-axis symmetry can be useful, as graphs that exhibit this type of symmetry mirror each other across the x-axis. This means if one point \((x, y)\) is on the graph, then \((x, -y)\) should also be on the graph if it were symmetric around the x-axis.

y-axis symmetry

Y-axis symmetry occurs when a graph looks the same on both sides of the y-axis. To check for this type of symmetry, we replace every \(x\) in the equation with \(-x\).

Let's apply this to our function \(x^{3}y = 1\). Replacing \(-x\) for \(x\) yields \((-x)^{3}y = -x^{3}y = 1\). This equation does not simplify to the original equation \(x^{3}y = 1\). Therefore, this function does not exhibit y-axis symmetry.

If a graph has y-axis symmetry, both sides of the graph are mirror images across the y-axis. In terms of coordinates, this implies that for every point \((x, y)\) on the graph, there is a corresponding point \((-x, y)\) if it were symmetric with respect to the y-axis.

origin symmetry

Origin symmetry is observed when a graph can be rotated 180 degrees around the origin (0,0) and remains unchanged. To test for this symmetry, replace \(x\) with \(-x\) and \(y\) with \(-y\) in the equation.

For our function \(x^{3}y = 1\), substituting \(-x\) for \(x\) and \(-y\) for \(y\) gives us \((-x)^{3}(-y) = x^{3}y = 1\). The simplified expression is identical to the original function, which confirms that it is symmetric with respect to the origin.

When a graph is symmetric about the origin, if a point \((x, y)\) lies on the graph, then the point \((-x, -y)\) also does. This can often suggest certain properties about the function, such as alternating signs in its function values.