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The problem involves analyzing the graph of the function \( f(x) = -4x^4 - x^3 + 4x^2 + x + 12 \) and determining its end behavior.

### Understanding the function:
- This is a **quartic function** because the highest degree term is \( -4x^4 \).
- The leading coefficient (the coefficient of \( x^4 \)) is **negative**, which will impact the end behavior.

### End Behavior of Polynomial Functions:
For large values of \( x \), the term with the highest degree (i.e., \( -4x^4 \)) dominates the behavior of the function. This term is negative, so:
1. As \( x \to \infty \), \( f(x) \to -\infty \).
2. As \( x \to -\infty \), \( f(x) \to -\infty \).

This implies that the end behavior of the function is:
- \( \lim_{x \to \infty} f(x) = -\infty \)
- \( \lim_{x \to -\infty} f(x) = -\infty \)

Thus, the correct option is:
- **\( \lim_{x \to -\infty} f(x) = -\infty \) and \( \lim_{x \to \infty} f(x) = -\infty \)**

Would you like further details or explanation on how to determine end behavior?

Here are 5 related questions you might want to consider:
1. How does the sign of the leading coefficient affect the end behavior of a polynomial?
2. What is the general behavior of a quartic function with a positive leading coefficient?
3. How can you determine turning points from the graph of a polynomial?
4. How does the degree of a polynomial influence its symmetry?
5. How do local extrema (like maximum and minimum points) relate to the derivative of a function?

**Tip**: For end behavior, always look at the leading term of the polynomial as \( x \to \infty \) or \( x \to -\infty \); it dictates the function's long-term behavior.

The graph of f(x) = -4x^4 - x^3 + 4x^2 + x + 12 is given. Determine the end behavior of f(x).

Learn how to determine the end behavior of the quartic function f(x) = -4x^4 - x^3 + 4x^2 + x + 12. Understand the influence of the leading term and how to predict the behavior as x approaches infinity and negative infinity. Suitable for high school students in Grades 10-12.

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