Question

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

Explanation:

Step1: Determine the leading term

The leading term of the polynomial \(f(x)=0.25x^{3}-x^{2}-1\) is \(0.25x^{3}\).

Step2: Analyze the degree and leading coefficient

The degree \(n = 3\) (odd) and the leading coefficient \(a=0.25>0\).
When \(x\to+\infty\), for the term \(y = ax^{n}\) with \(n\) odd and \(a>0\), \(y = 0.25x^{3}\to+\infty\).
When \(x\to-\infty\), for the term \(y=ax^{n}\) with \(n\) odd and \(a > 0\), \(y=0.25x^{3}\to-\infty\).

Answer:

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