https://file.aiquickdraw.com/mathbot/file/1728865067510ekomelgv.jpg

The given polynomial function is:

\[
P(x) = -x^6 + x^5 - x^4 + 2
\]

### Analyzing the End Behavior:
For end behavior, we focus on the highest degree term, which in this case is \(-x^6\). 

1. **Leading Term**: The degree of the polynomial is 6, which is an even number. The leading coefficient is \(-1\), a negative number.

2. **End Behavior Based on Degree and Leading Coefficient**:
   - Since the degree is even and the leading coefficient is negative, the graph will behave as follows:
     - As \(x \to \infty\), \(P(x) \to -\infty\).
     - As \(x \to -\infty\), \(P(x) \to -\infty\).
   
   Therefore, the graph falls on both sides (left and right).

### Conclusion:
The correct sketch should show the graph falling towards \(-\infty\) on both the left and right ends.

Let me know if you have any questions or want further details!

Here are 5 questions that could expand this discussion:
1. How do the other terms in the polynomial affect the overall shape of the graph?
2. What happens at the turning points of the graph for a polynomial of degree 6?
3. Can you explain how the degree of a polynomial affects its end behavior?
4. How can we find the x-intercepts of this polynomial function?
5. How do changes in the leading coefficient influence the end behavior of any polynomial?

**Tip:** For end behavior analysis, always focus on the highest-degree term as it dominates the behavior of the function at extreme values of \(x\).

Select one of the four sketches (A)-(D) that follow to describe the end behavior of the graph of the function P(x) = -x^6 + x^5 - x^4 + 2.

Understand the end behavior of the polynomial function P(x) = -x^6 + x^5 - x^4 + 2 by focusing on the leading term -x^6. Learn how to analyze the behavior as x approaches infinity and negative infinity. Suitable for students in Grades 11-12.

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