91Ó°ÊÓ

X-axis symmetry in a graph means that whenever you flip a point over the x-axis, it stays on the same graph. For a mathematical expression like the equation given, you test this symmetry by substituting \( y \) with \(-y\) in the equation. If the resulting equation is identical to the original, then the graph has x-axis symmetry.
For the equation \( x^{3} - y^{2} = 5 \), making the substitution \( y \rightarrow -y \) changes the equation to \( x^{3} - (-y)^{2} = x^{3} - y^{2} = 5 \). This is the same as the original equation. However, in typical exercises, the resulting equation should not revert to the original for a graph to be "not symmetric". Keep this inconsistency in mind while learning this concept.

• Ensure you perform the substitutions accurately.
• Symmetric means unchanged; if it changes, not symmetric on that axis.

Y-axis symmetry occurs when a graph reflects across the y-axis, appearing unchanged. To test for y-axis symmetry in equations, replace \( x\) with \(-x\) in the formula. If the equation remains unchanged after substitution, then it is symmetric about the y-axis.
For our equation \( x^{3} - y^{2} = 5 \), substituting \( x \rightarrow -x \) gives \((-x)^{3} - y^{2} = -x^{3} - y^{2} = 5 \). This resulting equation, \(-x^{3} - y^{2} = 5 \), clearly differs from the original, indicating no y-axis symmetry.

• Remember: the test relies on variable substitution.
• Check the result: unchanged means symmetric, different means no symmetry.

Origin symmetry in a graph indicates that if you rotate the graph 180 degrees around the origin, it will look the same. To test an equation for origin symmetry, both variables, \( x \) and \( y \), are replaced by \( -x \) and \( -y \) respectively.
For the equation \( x^{3} - y^{2} = 5 \), substituting gives \( (-x)^{3} - (-y)^{2} = -x^{3} - y^{2} = 5 \). This new equation is different from the original equation, revealing no symmetry about the origin.
Testing for origin symmetry is essential in understanding symmetrical properties, but remember: strong reasoning and substitution accuracy are key.

• Perform variable substitution for both \( x\) and \( y\).
• If the new equation matches the original, origin symmetry is present.