- The end behavior is determined by the leading term \(x^5\), giving \(f(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\) and \(f(x) \rightarrow +\infty\) as \(x \rightarrow +\infty\).
- Factoring gives \(f(x) = x^3(x-4)^2\), identifying roots at \(x=0\) (multiplicity 3) and \(x=4\) (multiplicity 2).
- The graph crosses the x-axis at \(x=0\) (odd multiplicity) and touches, but does not cross, the x-axis at \(x=4\) (even multiplicity).
- Final answers: End behavior is \(f(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\) and \(f(x) \rightarrow +\infty\) as \(x \rightarrow +\infty\); the graph touches the x-axis at \(x=\boxed{4}\); the graph crosses the x-axis at \(x=0\).
Explanation
- Analyzing the Function and Factoring
We are given the polynomial function \(f(x) = x^5 - 8x^4 + 16x^3\). Our objective is to determine its end behavior, identify where the graph touches the x-axis, and describe its behavior at its roots.
First, we can factor the function to find its roots:
\[f(x) = x^3(x^2 - 8x + 16)\]
The expression inside the parentheses is a perfect square trinomial:
\[x^2 - 8x + 16 = (x-4)^2\]
Thus, the factored form of the function is:
\[f(x) = x^3(x-4)^2\]
- Determining End Behavior
To find the end behavior of the polynomial \(f(x) = x^5 - 8x^4 + 16x^3\), we examine the leading term, which is \(x^5\).
As \(x \rightarrow +\infty\), \(x^5 \rightarrow +\infty\), so \(f(x) \rightarrow +\infty\).
As \(x \rightarrow -\infty\), \(x^5 \rightarrow -\infty\) (because the exponent 5 is odd), so \(f(x) \rightarrow -\infty\).
Therefore, the end behavior is \(f(x) \rightarrow -\infty\) as \(x \rightarrow -\infty\) and \(f(x) \rightarrow +\infty\) as \(x \rightarrow +\infty\).
- Identifying Roots and X-axis Behavior
From the factored form \(f(x) = x^3(x-4)^2\), we can identify the roots:
- The root at \(x=0\) has a multiplicity of 3 (an odd number).
- The root at \(x=4\) has a multiplicity of 2 (an even number).
According to the rules of multiplicities:
- If a root has an odd multiplicity, the graph crosses the x-axis at that root. Since the multiplicity of \(x=0\) is 3, the graph crosses the x-axis at \(x=0\).
- If a root has an even multiplicity, the graph touches the x-axis and turns around at that root. Since the multiplicity of \(x=4\) is 2, the graph touches, but does not cross, the x-axis at \(x=4\).
- Conclusion
The end behavior is \(f(x) \rightarrow -\infty \text{ as } x \rightarrow -\infty\) and \(f(x) \rightarrow +\infty \text{ as } x \rightarrow +\infty\).
The graph touches, but does not cross, the x-axis at \(x = 4\).
The graph of the function crosses the x-axis at \(x = 0\).
Examples
The behavior of polynomial graphs is crucial in fields like economics or engineering. For instance, if a company's profit is modeled by a polynomial, finding where the graph crosses the x-axis tells us the "break-even points," where the profit is zero. Understanding whether a graph "touches" or "crosses" at these points helps determine if profit shifts from negative to positive or just touches zero and stays the same. Just like in our function \(f(x) = x^3(x-4)^2\), understanding these key points allows us to map out important performance indicators.