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Let's work through the problem you've shared, which involves analyzing the polynomial function \( f(x) = x^3 - 4x^2 - x + 4 \). 

### Part (a) - End Behavior:
You're asked to determine the end behavior of the function using the leading coefficient test. This function has a degree of 3, which is odd, and the leading coefficient of \(x^3\) is positive (1). For a cubic function with a positive leading coefficient, the end behavior will be:

- The graph falls to the left and rises to the right.

This matches the answer marked as **A: The graph falls to the left and rises to the right**.

### Part (b) - X-Intercepts:
To find the x-intercepts, we need to solve \( f(x) = 0 \), which gives the roots of the polynomial:

\[ x^3 - 4x^2 - x + 4 = 0 \]

The given x-intercepts are \( x = 1, 4, -1 \), which means these are the points where the graph touches or crosses the x-axis. At each intercept:
- At \( x = 1 \), the graph crosses the x-axis.
- At \( x = 4 \), the graph crosses the x-axis.
- At \( x = -1 \), the graph crosses the x-axis.

Since the polynomial is degree 3, it must cross the x-axis at these points because odd-degree functions always cross the x-axis at least once.

### Part (c) - Does the Graph Cross at Specific X-Intercepts?
The question seems to ask whether the graph crosses the x-axis at specific intercepts or simply touches it. Since the graph crosses the x-axis at all three given x-values, the answer to the selection part **B** ("There are no x-intercepts where the graph does not cross the x-axis") would be **false**. 

Would you like more detailed explanations or to go further into the graphing analysis?

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Here are 5 follow-up questions related to this material:

1. How does the degree of a polynomial affect its end behavior?
2. What happens to the graph of a cubic function when the leading coefficient is negative?
3. How can you find whether a function touches or crosses the x-axis at a root?
4. Can a cubic function have only one x-intercept? Explain why or why not.
5. What does the sign of the leading term indicate about the direction of the graph?

**Tip:** When solving polynomial equations, factorization or the Rational Root Theorem can help find x-intercepts.

Analyze the polynomial function f(x) = x^3 - 4x^2 - x + 4. Use the leading coefficient test to determine the end behavior and find the x-intercepts.

Learn how to determine the end behavior of the cubic polynomial f(x) = x^3 - 4x^2 - x + 4 using the leading coefficient test, and find its x-intercepts. This topic covers polynomial function analysis, end behavior, and solving polynomial equations.

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