Problem 78 - Algebra and Trigonometry

Question

\(77-82\) . Test the equation for symmetry. $$ x=y^{4}-y^{2} $$

Step-by-Step Solution

Answer

The equation is symmetric with respect to the x-axis.

Step 1: Identify Symmetry Types

There are three main types of symmetry to test for: 1. Symmetry with respect to the x-axis. 2. Symmetry with respect to the y-axis. 3. Symmetry with respect to the origin.

Step 2: Test for X-axis Symmetry

Replace \(y\) with \(-y\) in the equation and simplify:\[x = (-y)^4 - (-y)^2\]This simplifies to:\[x = y^4 - y^2\]The equation remains the same, so the function is symmetric with respect to the x-axis.

Step 3: Test for Y-axis Symmetry

Replace \(x\) with \(-x\) in the equation and solve for \(y\):\[-x = y^4 - y^2\]This does not simplify to the original equation, so there is no symmetry with respect to the y-axis.

Step 4: Test for Origin Symmetry

Replace \(x\) with \(-x\) and \(y\) with \(-y\):\[-x = (-y)^4 - (-y)^2\]This simplifies to:\[-x = y^4 - y^2\]This does not match the original equation, indicating that there is no symmetry with respect to the origin.

Key Concepts

X-axis Symmetry

Understanding x-axis symmetry in mathematical equations is important because it reveals specific characteristics of a function's graph. An equation is symmetric with respect to the x-axis when replacing

• \(y\) with \(-y\)

in the equation leaves it unchanged. This means the graph of the function reflects itself across the x-axis.
For example:
Consider the equation \(x = y^4 - y^2\). When you substitute \(-y\) for \(y\), the equation becomes \(x = (-y)^4 - (-y)^2\). Simplifying it, you get:
\[x = y^4 - y^2\]
Since this is the original equation, it confirms x-axis symmetry.
In practical terms:

• Any point \((x, y)\) on the graph corresponds to a point \((x, -y)\) as well.

Y-axis Symmetry

Y-axis symmetry in equations can help identify if a function is mirrored across the y-axis. An equation is symmetric with respect to the y-axis if replacing

• \(x\) by \(-x\)

results in an equivalent equation. It's like folding the graph over the y-axis and finding that both sides match.
However, for our example, \(x = y^4 - y^2\), substituting \(-x\) for \(x\) makes the equation \(-x = y^4 - y^2\).
This transformation does not simplify back to the original equation, so the function does not demonstrate y-axis symmetry. Knowing this:

• A point \((x, y)\) does not necessarily have a counterpart at \((-x, y)\) on the graph.

Origin Symmetry

Origin symmetry is a fascinating aspect of mathematical functions where the graph appears identical when rotated 180 degrees around the origin. An equation is symmetric with respect to the origin when replacing:

• \(x\) with \(-x\) and
• \(y\) with \(-y\)

results in the same equation.
For example:
In the case of \(x = y^4 - y^2\), substituting \(-x\) for \(x\) and \(-y\) for \(y\), the equation modifies to \(-x = (-y)^4 - (-y)^2\). Simplifying gives:
\[-x = y^4 - y^2\]
This doesn't match the original equation, indicating a lack of origin symmetry. Consequently, no point \((x, y)\) on its graph has a reflected point \((-x, -y)\). This highlights how certain transformations can reveal the inherent symmetry in equations, or the lack thereof.