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Imagine the x-axis as a mirror lying flat. To determine if a graph is symmetrical along this line, we perform a mini-test. Replace every instance of \( y \) with \( -y \) in your equation. If the resulting equation looks identical to the original, you’ve got x-axis symmetry!
For example, with our equation \( x = |y| \), if we swap \( y \) for \( -y \), it becomes \( x = |-y| \). Here's the neat part: absolute values embrace both positive and negative equally, so \( |-y| = |y| \). Nothing changes!
This means that flipping over the x-axis doesn’t alter our graph. Hence, it’s symmetrical along the x-axis. Keep this handy trick in mind: if flipping the \( y \) value results in no net change, x-axis symmetry is confirmed.
It's useful to verify this visually, too. Imagine folding your graph along the x-axis. If both sides align perfectly, you’re looking at a graph with true x-axis symmetry.

• Replace \( y \) with \( -y \)
• Check if equation remains unchanged
• If unchanged, x-axis symmetry is confirmed

The y-axis is a vertical line that splits the graph into left and right halves. Checking for y-axis symmetry involves rewriting the equation by swapping \( x \) with \( -x \). The test is simple: if this transformation doesn't affect the structure of the equation, you've got symmetry.
Take the equation \( x = |y| \). By replacing \( x \) with \( -x \), we get \( -x = |y| \). This results in a different equation than our start, \( x = |y| \), showing a lack of symmetry about the y-axis.
Why does this matter? It's about balance! For true y-axis symmetry, folding your graph along the y-axis should result in perfect overlap. This isn’t the case here, which confirms the absence of symmetry about the y-axis.
Remember this step-by-step method to quickly test an equation for y-axis symmetry:

• Replace \( x \) with \( -x \)
• If the equation is unchanged, y-axis symmetry exists
• No change means the graph has y-axis symmetry

Origin symmetry might seem like a trick - it requires flipping two things at once! For this, both \( x \) and \( y \) change to \( -x \) and \( -y \), a full 180-degree turnaround from any point on the graph. To see if the equation \( x = |y| \) is symmetrical about the origin, replace both \( x \) with \( -x \) and \( y \) with \( -y \): \[-x = |-y|\].Because absolutes remain unchanged, \[-x = |y|\] is the transformed equation. Compare this with the original, \( x = |y| \). Since they're not identical, origin symmetry doesn't occur. In essence, for origin symmetry to hold, post-transforming both variables shouldn’t disrupt the equation's balance. Visualize twisting your graph around the origin. A perfect match on both sides confirms symmetry.

• Change \( x, y \) to \( -x, -y \)
• Check for unchanged equation structure
• Dissimilar equations mean no origin symmetry