In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.

𝔂 = x² - 2x - 1

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To determine symmetry about the y-axis, check if the function is even. A function f(x) is even if f(x) = f(-x) for all x in the domain. Substitute -x into the function: y = (-x)² - 2(-x) - 1.

Simplify the expression: y = x² + 2x - 1. Compare this with the original function y = x² - 2x - 1. Since they are not equal, the function is not symmetric about the y-axis.

To determine symmetry about the origin, check if the function is odd. A function f(x) is odd if -f(x) = f(-x) for all x in the domain. Substitute -x into the function: y = (-x)² - 2(-x) - 1 and simplify to get y = x² + 2x - 1.

Now, check if -f(x) = f(-x). Calculate -f(x): -f(x) = -(x² - 2x - 1) = -x² + 2x + 1. Compare this with f(-x) = x² + 2x - 1. Since they are not equal, the function is not symmetric about the origin.

Since the function is neither even nor odd, it is neither symmetric about the y-axis nor the origin.

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Symmetry about the y-axis

A function is symmetric about the y-axis if replacing x with -x in the function yields the same function. Mathematically, this means that f(-x) = f(x). For polynomial functions, this often indicates that all the exponents of x are even.

Symmetry about the origin

A function is symmetric about the origin if replacing x with -x and y with -y results in the same function. This is expressed as f(-x) = -f(x). Functions that exhibit this symmetry typically contain only odd powers of x.

Analyzing polynomial functions

Polynomial functions can be analyzed for symmetry by examining their terms. The degree and the coefficients of the terms determine the function's behavior under transformations. For the given function, identifying the terms will help in determining its symmetry properties.

In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.𝔂 = x² - 2x - 1