Question

1. writing explain what is meant by the end behavior of a polynomial function.

Explanation:

The end behavior of a polynomial function describes how the graph of the function behaves as \(x\) approaches positive infinity (\(x\to+\infty\)) and negative infinity (\(x\to-\infty\)). It is determined by the leading term (the term with the highest degree) of the polynomial. For a polynomial function \(f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\) (\(n\geq0\), \(a_n
eq0\)), as \(x\to\pm\infty\), \(f(x)\approx a_nx^n\). If \(n\) is even: when \(a_n>0\), \(f(x)\to+\infty\) as \(x\to\pm\infty\); when \(a_n < 0\), \(f(x)\to-\infty\) as \(x\to\pm\infty\). If \(n\) is odd: when \(a_n>0\), \(f(x)\to+\infty\) as \(x\to+\infty\) and \(f(x)\to-\infty\) as \(x\to-\infty\); when \(a_n < 0\), \(f(x)\to-\infty\) as \(x\to+\infty\) and \(f(x)\to+\infty\) as \(x\to-\infty\).

Answer:

The end behavior of a polynomial function describes the behavior of the function's graph as \(x\) approaches positive and negative infinity, which is mainly determined by the leading term (highest - degree term) of the polynomial.