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Y-axis symmetry is one of the simplest forms of symmetry to understand when it comes to equations. If you reflect a graph across the y-axis and it remains unchanged, then it is said to have y-axis symmetry. To test whether an equation has this type of symmetry, you replace each occurrence of the variable \( x \) with \( -x \). Here's what you need to do:\

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• Substitute \( -x \) for every \( x \) in the equation.
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• Simplify the equation.
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• If the resulting equation is equivalent to the original one, it has y-axis symmetry.
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\For instance, in our example, substituting \( -x \) for \( x \) in \( x^{4} y^{4}+x^{2} y^{2}=1 \) still results in the same equation, confirming its y-axis symmetry. This is often easier to spot in equations involving even powers of \( x \), since raising a negative number to an even power yields the same result as raising its positive counterpart.

Understanding x-axis symmetry involves a similar process, but with respect to the x-axis instead. It's a little less intuitive because it involves the transformation of the variable \( y \). Here's how you can test for x-axis symmetry:\

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• Replace each occurrence of the variable \( y \) with \( -y \).
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• Simplify the resulting equation.
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• If the equation remains unchanged, it indicates x-axis symmetry.
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\In the case of our example equation \( x^{4} y^{4}+x^{2} y^{2}=1 \), replacing \( y \) with \( -y \) leads to the original equation once again. This demonstrates that the equation is symmetric with respect to the x-axis. It's important to note that x-axis symmetry is often associated with even powers of \( y \), since any negative value raised to an even power will result in a positive value.

Origin symmetry might seem complex, but it's very logical at its core. For an equation to be symmetric about the origin, a graph should remain unchanged if rotated 180 degrees around the origin. This involves changes in signs for both variables \( x \) and \( y \). Here's what you need to do:\

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• Substitute \( -x \) for \( x \) and \( -y \) for \( y \) in the equation.
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• Simplify as before.
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• If the resulting equation matches the original, the equation has origin symmetry.
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\In our example equation, performing this double substitution still results in the equation \( x^{4} y^{4}+x^{2} y^{2}=1 \). This clearly shows that it is symmetric about the origin. Generally, origin symmetry is seen in equations where both \( x \) and \( y \) are raised to even powers, ensuring that the negativity of \(-x\) and \(-y\) cancel out the change in sign. This kind of symmetry is a strong indicator of a balanced and harmonious property inherent in the equation.