To analyze the end behavior of the polynomial \( f(x) = x^4 - 2x^3 - 3x^2 \), we examine the behavior of the function as \( x \to \infty \) and \( x \to -\infty \).

### End Behavior as \( x \to \infty \):
1. The leading term of the polynomial is \( x^4 \), which dominates the behavior of the function as \( x \) becomes very large.
2. Since the degree of the leading term \( x^4 \) is even and the coefficient is positive, the function will approach \( +\infty \) as \( x \to \infty \).

Thus:
- As \( x \to \infty \), \( f(x) \to +\infty \).

### End Behavior as \( x \to -\infty \):
1. Similarly, when \( x \) becomes very negative, the leading term \( x^4 \) still dominates.
2. Since \( x^4 \) is an even-powered term, regardless of whether \( x \) is positive or negative, \( x^4 \) will be positive.

Thus:
- As \( x \to -\infty \), \( f(x) \to +\infty \).

### Conclusion:
- The end behavior of \( f(x) = x^4 - 2x^3 - 3x^2 \) is:
  - As \( x \to \infty \), \( f(x) \to +\infty \).
  - As \( x \to -\infty \), \( f(x) \to +\infty \).

Do you want more details or have any questions?

---

Here are some related questions:
1. What is the degree and leading coefficient of a polynomial?
2. How does the end behavior change with different leading terms?
3. What are critical points and how are they determined?
4. How does the behavior of odd-degree polynomials differ from even-degree polynomials?
5. Can you explain the role of non-leading terms in the overall shape of the graph?

**Tip:** The end behavior of a polynomial is always determined by its leading term.

Explore the end behavior of the polynomial x^4 - 2x^3 - 3x^2. Understand how the leading term determines the function's behavior as x approaches infinity and negative infinity. Ideal for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation