- The end behavior of the polynomial \(f(x) = x^5 - 8x^4 + 16x^3\) is determined by its leading term, \(x^5\).
- Since the degree \(n=5\) is odd and the leading coefficient \(a_5=1\) is positive, the function behaves like \(y=x^5\) for large \(|x|\).
- As \(x \to +\infty\), \(f(x) \to +\infty\).
- As \(x \to -\infty\), \(f(x) \to -\infty\).
- The correct end behavior is \(f(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\); \(f(x) \rightarrow +\infty\) as \(x \rightarrow +\infty\).
- \(\boxed{f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty; f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty}\)
Explanation
- Identifying the Leading Term
The polynomial function is given by \(f(x) = x^5 - 8x^4 + 16x^3\). To determine its end behavior, we look at the leading term of the polynomial, which is the term with the highest power of \(x\). In this case, the leading term is \(x^5\). The end behavior of a polynomial is determined solely by the behavior of its leading term as \(x\) approaches positive or negative infinity.
- Analyzing the Leading Term's Characteristics
The degree of the polynomial is \(n = 5\), which is an odd number. The leading coefficient is the coefficient of the \(x^5\) term, which is \(a_n = 1\). Because the leading coefficient is positive (\(a_n > 0\)) and the degree is odd, the polynomial will mimic the behavior of the parent power function \(y = x^n\) for very large values of \(|x|\). Specifically, for large \(x\), the \(x^5\) term dominates all other terms in the polynomial.
- Determining the End Behavior
As \(x \to +\infty\), the term \(x^5\) grows without bound toward positive infinity. Thus, the function \(f(x)\) approaches \(+\infty\).
As \(x \to -\infty\), since the exponent 5 is odd, \(x^5\) becomes a large negative number. As the magnitude of \(x\) increases, \(x^5\) approaches negative infinity. Thus, the function \(f(x)\) approaches \(-\infty\).
- Conclusion
Comparing our findings with the given options:
- \(f(x) \to -\infty\) as \(x \to -\infty\); \(f(x) \to +\infty\) as \(x \to +\infty\).
This matches the second option provided in the problem. Thus, the correct description of the end behavior is \(f(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\) and \(f(x) \rightarrow +\infty\) as \(x \rightarrow +\infty\).
Examples
Understanding the end behavior of a polynomial is crucial in fields like economics or physics when modeling long-term trends. For example, if a company's profit over time is modeled by a polynomial function, knowing the end behavior tells stakeholders whether the profit will eventually grow indefinitely or decline toward a loss as time increases, even if the short-term fluctuations are complex. Similarly, in physics, it helps in predicting the behavior of systems under extreme conditions where one variable becomes very large.