91Ó°ÊÓ

Symmetry with respect to the axes is a fascinating aspect of math. When examining a function's graph, you might wonder if it looks the same on both sides of an axis, giving it balance. For y-axis symmetry:

• Replace every instance of \(x\) with \(-x\) in the function equation.
• If the new equation \(f(-x)\) matches the original \(f(x)\), the graph is symmetric about the y-axis.

For example, consider the function \(y = x^2 - x\). By replacing \(x\) with \(-x\), we get \(f(-x) = x^2 + x\). Since \(x^2 + x eq x^2 - x\), this function isn't symmetric about the y-axis.

The x-axis symmetry applies mainly to relations rather than functions, since replacing \(y\) with \(-y\) is not standard for typical function analysis. It's more relevant when dealing with circles or other geometrical shapes. If needed, replace \(y\) with \(-y\) to check this symmetry for relations.

Origin symmetry occurs when a function looks the same when rotated 180 degrees around the origin. To check for it:

• Replace \(x\) with \(-x\) in the equation.
• Compare the new function \(f(-x)\) to \(-f(x)\) (the negation of the original function).

If \(f(-x) = -f(x)\), the graph has origin symmetry. In this case, the function \(y = x^2 - x\) transforms to \(f(-x) = x^2 + x\). Then, the negated function \(-f(x) = -x^2 + x\). Since these do not match, the function lacks origin symmetry. This type of symmetry can create visually appealing patterns and is common in certain algebraic structures.

Functions can be classified as even or odd based on their symmetry. An even function:

• Has y-axis symmetry.
• Satisfies \(f(-x) = f(x)\).

Think of functions like \(f(x) = x^2\), where replacing \(x\) with \(-x\) results in the same function. These functions have a mirror-like symmetry about the y-axis.

An odd function, on the other hand:

• Has origin symmetry.
• Satisfies \(f(-x) = -f(x)\).

For example, the function \(f(x) = x^3\) is odd. Here, \(f(-x) = -x^3\) matches \(-f(x)\). Our function \(y = x^2 - x\) is neither even nor odd because it doesn't satisfy either condition. Recognizing these properties can simplify integration and other mathematical operations.