Problem 46 - Algebra and Trigonometry

Question

45–50 ? Test the equation for symmetry. $$ x=y^{4}-y^{2} $$

Step-by-Step Solution

Answer

The equation is symmetric about the x-axis, not the y-axis or the origin.

Step 1: Identify the Symmetry Types

Before testing for symmetry, let's identify the types of symmetry we can check for: symmetry about the x-axis, y-axis, and origin.

Step 2: Test for Symmetry about the x-axis

To test for symmetry about the x-axis, replace every occurrence of with -y in the equation. If the equation remains unchanged, it is symmetric about thex-axis. Start with:\[ \begin{align*} x & = (y^{4}) - (y^{2}) \end{align*} \]Replace with , to get \[ \begin{align*} x & = ((-y)^{4}) - ((-y)^{2}) \end{align*} \]which simplifies to \[ \begin{align*} x &= y^{4} - y^{2}. \end{align*} \]The equation is the same so it is symmetric about the x-axis.

Step 3: Test for Symmetry about the y-axis

To test for symmetry about the y-axis, replace every x with -x in the equation \[ x = y^{4} - y^{2} \].We get the equation:\[ -x = y^{4} - y^{2} \].This is not equal to the original equation, so the equation is not symmetric about the y-axis.

Step 4: Test for Symmetry about the Origin

To test for symmetry about the origin, replace with -y and x with -x. The equation becomes:\[ -x = (-y)^{4} - (-y)^{2}. \]This simplifies to:\[ -x = y^{4} - y^{2}. \]Which is not equivalent to the original equation \[ x = y^{4} - y^{2} \], hence the equation is not symmetric about the origin.

Key Concepts

x-axis symmetry

When we talk about x-axis symmetry, we're talking about a reflection across the x-axis. This means that if you take any point on the graph and flip it over the x-axis, you'll land on another point on the graph. In simpler terms, the top half of the graph should mirror the bottom half.

To check if an equation is symmetric about the x-axis, replace every 'y' in the equation with '-y'. For example, in the equation \( x = y^{4} - y^{2} \), when we replace \( y \) with \( -y \), we get:

• The new equation is \( x = (-y)^{4} - (-y)^{2} \).
• Because even powers of \( y \) are unchanged by this sign change, \( (-y)^{4} = y^{4} \) and \((-y)^{2} = y^{2} \), leading us back to the original equation: \( x = y^{4} - y^{2} \).
• Since the equation remains the same, it is symmetric about the x-axis.

This type of symmetry is useful when predicting behavior of equations in relation to the x-axis.

y-axis symmetry

Symmetry about the y-axis implies flipping the graph over the y-axis results in the same graph. Think of it as the right side of the graph being a mirror image of the left side.

To test for y-axis symmetry, replace every \( x \) with \( -x \). If the equation remains unchanged, the graph is y-axis symmetric. For our equation \( x = y^{4} - y^{2} \), we substitute:

• Replace \( x \) with \( -x \) to get \( -x = y^{4} - y^{2} \).
• This equation \( -x = y^4 - y^2 \) is not the same as the original \( x = y^4 - y^2 \).
• Therefore, the equation is not symmetric about the y-axis.

Y-axis symmetry is useful for visualizing both sides of a graph around the vertical axis, but not all equations have this property.

origin symmetry

Origin symmetry involves a double reflection, where a point on the graph is flipped across both the x and y axes and still falls onto the graph. You can think of it as rotating the graph 180 degrees about the origin.

To determine if an equation has origin symmetry, replace \( x \) with \( -x \) and \( y \) with \( -y \) in the equation. For the original equation \( x = y^{4} - y^{2} \):

• Substitute and transform to get: \( -x = (-y)^{4} - (-y)^{2} \).
• This simplifies to \( -x = y^4 - y^2 \), which is not equivalent to the original equation \( x = y^4 - y^2 \).
• Thus, the equation does not possess origin symmetry.

Origin symmetry is significant for understanding how a graph behaves when flipped around the center point, but, as we see here, not all equations demonstrate this symmetry.