The image shows a math problem asking to "Determine the end behavior of $f(x) = x^3 - 2x^2 + x - 4$." Below are four multiple-choice options for the end behavior of the polynomial function.

To determine the end behavior of a polynomial, we focus on the leading term, which is the term with the highest degree. In this case, the leading term is $x^3$.

For $f(x) = x^3 - 2x^2 + x - 4$:

• The degree of the polynomial is 3 (odd degree).
• The coefficient of the leading term is positive (1).

The general rule for polynomials is:

• For an odd-degree polynomial with a positive leading coefficient, as $x \to +\infty$, $f(x) \to +\infty$ and as $x \to -\infty$, $f(x) \to -\infty$.

Thus, the end behavior is:

• As $x \to +\infty$, $f(x) \to +\infty$
• As $x \to -\infty$, $f(x) \to -\infty$

Now, you can select the option that matches this behavior.

Would you like further details on how this is calculated or any other assistance?

Here are 5 related questions to explore:

1. What are the end behaviors of even-degree polynomials with negative leading coefficients?
2. How do you find the degree and leading term of more complex polynomials?
3. Can the end behavior of a function be affected by non-leading terms?
4. What are some real-world applications of polynomial end behavior?
5. How can graphing a polynomial function help confirm the end behavior?

Tip: When analyzing polynomial functions, always start by identifying the leading term for insights on end behavior.