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Graphs that have y-axis symmetry look the same on both sides of the vertical (y) axis. This kind of symmetry implies that if you folded the graph along the y-axis, both sides would match perfectly.
To test for y-axis symmetry in an equation, you replace every instance of \(x\) with \(-x\). If the resulting equation is identical to the initial one, then the graph has symmetry with respect to the y-axis.
In mathematical terms, if \(f(x) = f(-x)\) for every \(x\) in the domain, the graph is symmetric about the y-axis. This is a hallmark of even functions.
For example, with the equation \(y^2 = x^2 - 2\), substituting \(-x\) gives \(y^2 = (-x)^2 - 2\), which simplifies back to \(y^2 = x^2 - 2\), confirming y-axis symmetry.

• To test for y-axis symmetry, swap \(x\) for \(-x\).
• Y-axis symmetry means the graph is mirrored over the vertical axis.
• This type of symmetry is seen in even functions.

X-axis symmetry in graphs means the picture is the same when flipped over the horizontal (x) axis. This type of symmetry implies that the top half of the graph is a mirror image of the bottom half.
To verify x-axis symmetry, replace \(y\) with \(-y\) in the equation. If the resulting equation matches the original, then the graph is symmetric about the x-axis.
For example, replace \(y\) with \(-y\) in the equation \(y^2 = x^2 - 2\), giving \((-y)^2 = x^2 - 2\), which simplifies to \(y^2 = x^2 - 2\). Thus, the original equation confirms x-axis symmetry.

• To check for x-axis symmetry, substitute \(y\) with \(-y\).
• Symmetry along the x-axis means the graph can fold over the horizontal and still match up.
• This property is not typical for function graphs as it implies it’s not a function if the graph includes points having the same x value but different y values.

Origin symmetry implies that every part of the graph has a matching counterpart through rotation of 180 degrees about the origin (0,0). In simple terms, if you rotate the graph half a turn, it looks the same.
To test for this type of symmetry, replace \(x\) with \(-x\) and \(y\) with \(-y\) concurrently in the equation. If you get back the original equation, the graph is symmetric with respect to the origin.
Looking at the equation \(y^2 = x^2 - 2\), if you substitute \(-x\) and \(-y\), the equation \((-y)^2 = (-x)^2 - 2\) simplifies back to \(y^2 = x^2 - 2\), confirming origin symmetry.

• To determine origin symmetry, shift \(x\) to \(-x\) and \(y\) to \(-y\) simultaneously.
• Origin symmetry means the graph looks the same if turned 180 degrees around the origin.
• This type of symmetry is typical in odd functions.