Math Problem Statement

Determine the end behavior of f(x) = x^3 - 2x^2 + x - 4.

Solution

The image shows a math problem asking to "Determine the end behavior of f(x)=x32x2+x4f(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 x3x^3.

For f(x)=x32x2+x4f(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+x \to +\infty, f(x)+f(x) \to +\infty and as xx \to -\infty, f(x)f(x) \to -\infty.

Thus, the end behavior is:

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

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
End Behavior
Leading Coefficient

Formulas

End behavior of odd-degree polynomials with positive leading coefficient

Theorems

End Behavior Theorem for Polynomials

Suitable Grade Level

Grades 9-12