Question

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

1. describe the end behavior of ( f(x) = 0.25x^3 - x^2 - 1 ).

Explanation:

The end - behavior of a polynomial function \(y = f(x)\) describes the trend of the graph of the function as \(x\to+\infty\) (as \(x\) gets very large positive) and as \(x\to-\infty\) (as \(x\) gets very large negative).

For a polynomial function \(f(x)=a_nx^n + a_{n - 1}x^{n-1}+\cdots+a_1x + a_0\), where \(n\) is a non - negative integer and \(a_n
eq0\), the end - behavior is determined by the leading term \(a_nx^n\).

For the function \(f(x)=0.25x^3 - x^2 - 1\), the leading term is \(0.25x^3\) (where \(n = 3\) (odd) and \(a_n=0.25>0\)).

As \(x\to+\infty\), \(y = f(x)\approx0.25x^3\). Since \(x^3\to+\infty\) as \(x\to+\infty\) and \(0.25>0\), \(y=f(x)\to+\infty\) as \(x\to+\infty\).

As \(x\to-\infty\), \(y = f(x)\approx0.25x^3\). Since \(x^3\to-\infty\) as \(x\to-\infty\) and \(0.25>0\), \(y = f(x)\to-\infty\) as \(x\to-\infty\).

Answer:

As \(x\to+\infty\), \(f(x)\to+\infty\); as \(x\to-\infty\), \(f(x)\to-\infty\).