The problem involves using the Leading Coefficient Test to determine the end behavior of the polynomial function:

$f(x) = -2x^4 + 7x^3 - x + 9$

Steps to apply the Leading Coefficient Test:

1. Identify the leading term: The leading term is the term with the highest degree in the polynomial. In this case, it's $-2x^4$, where:

   • The degree is 4 (an even number).
   • The leading coefficient is $-2$ (a negative number).

2. Determine end behavior based on degree and leading coefficient:

   • For a polynomial with an even degree (like 4), the behavior of the graph at the ends (as $x \to \infty$ and $x \to -\infty$) depends on the sign of the leading coefficient:
     • If the leading coefficient is positive, the graph rises to both ends.
     • If the leading coefficient is negative, the graph falls to both ends.

Since the leading coefficient is $-2$ (negative) and the degree is even, the graph will fall to the left and fall to the right.

Correct Answer:

The correct option is B: "The graph of f(x) falls to the left and falls to the right."

Would you like any more details or clarification on this? Here are some related questions to deepen your understanding:

1. How does the Leading Coefficient Test differ for odd-degree polynomials?
2. What happens if the degree is even but the leading coefficient is positive?
3. How does the graph change if you modify the leading coefficient of this function to a positive number?
4. What role do the other terms in the polynomial (like $7x^3$ and $-x$) play in determining the shape of the graph?
5. How can you find the turning points of a polynomial function like this one?

Tip: For polynomial functions, the degree tells you about the overall shape of the graph, while the leading coefficient helps determine the end behavior.