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?

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.