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The end behavior of a function describes how the function behaves (what the function values approach) as the independent variable, often denoted as x, approaches positive infinity (\(+\infty\)) and negative infinity (\(-\infty\)). Essentially, it's studying where the function heads as the values move to the far left and far right on a graph.

For polynomial functions, the end behavior is determined by the degree and the sign of the leading coefficient. The degree of the polynomial is the highest power of the variable, and the leading coefficient is the coefficient of the term with the highest degree.

Here are the general rules for end behavior of polynomial functions: \n\n1. If the degree of the polynomial is even, and the leading coefficient is positive, the end behavior is: as \(x \to +\infty, f(x) \to +\infty\) and as \(x \to -\infty, f(x) \to +\infty\). Meaning the graph rises to the left and rises to the right. \n\n2. If the degree of the polynomial is even, and the leading coefficient is negative, the end behavior is: as \(x \to +\infty, f(x) \to -\infty\) and as \(x \to -\infty, f(x) \to -\infty\). Meaning the graph falls to the left and falls to the right. \n\n3. If the degree of the polynomial is odd, and the leading coefficient is positive, the end behavior is: as \(x \to +\infty, f(x) \to +\infty\) and as \(x \to -\infty, f(x) \to -\infty\). Meaning the graph falls to the left and rises to the right. \n\n4. If the degree of the polynomial is odd, and the leading coefficient is negative, the end behavior is: as \(x \to +\infty, f(x) \to -\infty\) and as \(x \to -\infty, f(x) \to +\infty\). Meaning the graph rises to the left and falls to the right.