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Understanding function symmetry with respect to the y-axis involves examining whether a function remains unchanged when you replace every instance of variable \(x\) with its opposite \(-x\). For a function \(y = f(x)\), this test determines if \(f(-x) = f(x)\). If the result is equal, the graph is symmetric about the y-axis. This means that any point \((x, y)\) on the graph will have a mirror image point \((-x, y)\) on the graph as well.

In our original function \(y = x^2 - x\), when we substitute \(x\) with \(-x\), we get \(y = x^2 + x\). Notice that this new equation differs from the original, confirming that the function lacks symmetry with respect to the y-axis. This lack of symmetry implies the graph of this function does not mirror across the y-axis.

If a function does exhibit this kind of symmetry, it typically suggests that the graph is even, meaning it has balanced behavior relative to the y-axis.

Checking for symmetry with respect to the x-axis involves a different procedure, focusing on flipping the signs of the \(y\)-coordinates in a function. For a function \(y = f(x)\), we replace \(y\) with \(-y\) and check if \(-y = f(x)\) mirrors the original equation. If it does, the graph is symmetric with respect to the x-axis. This reveals that any point \((x, y)\) on the graph also has a corresponding point \((x, -y)\).

However, our test function \(y = x^2 - x\) did not meet this criterion. Substituting \(-y\) in place of \(y\) results in the equation \(-y = x^2 - x\), which does not equate to the original function. Consequently, this function is not symmetrical with respect to the x-axis.

Usually, symmetry about the x-axis applies to relations that may not necessarily represent functions, as a genuine function can only map one \(y\)-value for each \(x\)-input.

A function is symmetric with respect to the origin if swapping both the \(x\) and \(y\) variables with their opposites \(-x\) and \(-y\) results in the same equation. Mathematically, for a function \(y = f(x)\), checking symmetry involves testing if \(-y = f(-x)\). If this holds true, the function's graph will display rotational symmetry about the origin, showing that any point \((x, y)\) has a mirror image at \((-x, -y)\).

In the case of our function \(y = x^2 - x\), swapping both \(x\) and \(y\) yields the equation \(-y = x^2 + x\). Observing the distinct differences between this and the original function equation, we conclude that \(y = x^2 - x\) is not symmetric around the origin.

Functions that are symmetric with respect to the origin are termed odd functions and reflect a unique balance, producing equivalent, yet opposite, outputs for opposite inputs.