https://file.aiquickdraw.com/mathbot/file/17284209735791oy17tqe.jpg

The problem asks to determine the end behavior of the polynomial:

\[
P(x) = -7x^6 + x^5 - x^4 - x + 6
\]

### Step 1: Identify the Leading Term
The term with the highest degree dictates the end behavior of a polynomial. Here, the leading term is:

\[
-7x^6
\]

This is a degree 6 polynomial with a negative leading coefficient.

### Step 2: End Behavior Analysis
For polynomials, the degree and the sign of the leading coefficient determine how the graph behaves as \( x \) approaches \( +\infty \) (positive infinity) and \( -\infty \) (negative infinity). In this case:

- **Even Degree (6th degree):** Both ends of the graph will move in the same direction.
- **Negative Leading Coefficient (-7):** Since the coefficient is negative, both ends of the graph will point downward.

### End Behavior Summary
- As \( x \to +\infty \), \( P(x) \to -\infty \).
- As \( x \to -\infty \), \( P(x) \to -\infty \).

### Conclusion
The end behavior resembles the downward curve in both directions. Looking at the provided options (A, B, C, D), the best match is **D**, which shows both ends of the graph pointing down.

Would you like more details or have any other questions?

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Here are 5 related questions for further exploration:
1. How does the degree of a polynomial affect its end behavior?
2. What role does the sign of the leading coefficient play in determining end behavior?
3. How do odd-degree polynomials differ in end behavior from even-degree polynomials?
4. What effect do lower-degree terms (e.g., \(x^5, x^4\)) have on the overall shape of the graph?
5. Can you describe how the graph would look if the leading term were \(+7x^6\) instead of \(-7x^6\)?

**Tip:** Always focus on the highest-degree term for end behavior analysis, as lower-degree terms only affect the local behavior.

Use an end behavior diagram to describe the end behavior of the given function: P(x) = -7x^6 + x^5 - x^4 - x + 6

Analyze the end behavior of the polynomial P(x) = -7x^6 + x^5 - x^4 - x + 6. Learn how the degree and leading coefficient determine how the graph behaves as x approaches infinity. Suitable for high school students in grades 10-12.

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