https://file.aiquickdraw.com/mathbot/file/1726839783106qjy2fui5.jpg

We are given the polynomial function \( p(x) = -2x^3 + 3x^2 + 10x - 15 \). The question asks for the correct statement about the behavior of the graph as \( x \to \infty \) (positive infinity) and \( x \to -\infty \) (negative infinity).

### Step 1: Analyze the end behavior
The leading term of the polynomial is \( -2x^3 \), which is the term with the highest power of \( x \) (degree 3). This term primarily determines the end behavior of the polynomial.

- When \( x \to \infty \) (as \( x \) increases without bound):
  - \( -2x^3 \to -\infty \) because the cubic term is negative.
  - Therefore, \( p(x) \to -\infty \).

- When \( x \to -\infty \) (as \( x \) decreases without bound):
  - \( -2x^3 \to \infty \), because when you cube a negative number, it remains negative, and multiplying by the negative coefficient gives a positive result.
  - Therefore, \( p(x) \to \infty \).

### Conclusion:
- As \( x \to \infty \), \( p(x) \to -\infty \).
- As \( x \to -\infty \), \( p(x) \to \infty \).

This matches the third option:
- **As \( x \to \infty \), \( p(x) \to -\infty \) and as \( x \to -\infty \), \( p(x) \to \infty \)**.

Let me know if you want further details or have any questions! Here are five related questions:

1. How does the degree of a polynomial affect its end behavior?
2. What happens to a polynomial's end behavior when the leading coefficient changes sign?
3. How does the graph of a cubic polynomial differ from that of a quadratic one?
4. What is the significance of the leading term in a polynomial's behavior?
5. How can we find turning points in a cubic polynomial?

**Tip:** The end behavior of a polynomial is determined solely by its highest-degree term. The other terms influence the shape near the origin or turning points, but not at infinity.

The polynomial −2x^3 + 3x^2 + 10x − 15 is graphed in the coordinate plane. Which statement about the graph is correct?

Explore the end behavior of the cubic polynomial −2x^3 + 3x^2 + 10x − 15 and find out how it behaves as x approaches positive and negative infinity. This problem helps students understand key concepts in polynomial graphing and end behavior. Suitable 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