How does the degree of a polynomial affect its end behavior?

The degree of a polynomial significantly influences its **end behavior**, which describes how the polynomial behaves as \( x \to \infty \) or \( x \to -\infty \). Here are the key points:

### 1. **Degree and Leading Term**:
   - The degree of a polynomial is the highest power of \( x \) in the polynomial.
   - The **leading term** (the term with the highest degree) dominates the polynomial's behavior for large values of \( |x| \). The degree and the coefficient of this term determine the end behavior.

### 2. **Odd-Degree Polynomials**:
   - For odd-degree polynomials, the ends of the graph go in **opposite directions**.
   - If the leading coefficient is positive, as \( x \to \infty \), \( f(x) \to \infty \), and as \( x \to -\infty \), \( f(x) \to -\infty \).
   - If the leading coefficient is negative, the directions are reversed: as \( x \to \infty \), \( f(x) \to -\infty \), and as \( x \to -\infty \), \( f(x) \to \infty \).

### 3. **Even-Degree Polynomials**:
   - For even-degree polynomials, the ends of the graph go in the **same direction**.
   - If the leading coefficient is positive, as \( x \to \pm\infty \), \( f(x) \to \infty \).
   - If the leading coefficient is negative, as \( x \to \pm\infty \), \( f(x) \to -\infty \).

### 4. **Magnitude of the Degree**:
   - While the degree determines the general shape of the graph (e.g., linear for degree 1, parabolic for degree 2), the **end behavior** depends only on the sign of the leading coefficient and whether the degree is odd or even.

### 5. **Graph Symmetry**:
   - Polynomials with **even degrees** often have symmetry about the \( y \)-axis or no symmetry at all.
   - Polynomials with **odd degrees** often have rotational symmetry about the origin.

### Summary Table:
| Degree (n) | Leading Coefficient > 0       | Leading Coefficient < 0      |
|------------|-------------------------------|-------------------------------|
| Even       | Both ends \( \to \infty \)    | Both ends \( \to -\infty \)   |
| Odd        | Left \( \to -\infty \), Right \( \to \infty \) | Left \( \to \infty \), Right \( \to -\infty \) |

Would you like examples or further clarification on any of these points?

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### Expanding Questions:
1. How does the end behavior change for polynomials with higher degrees (e.g., \( n = 5, n = 6 \))?
2. How can you determine the leading coefficient of a polynomial if given a graph?
3. What are some real-world applications of understanding polynomial end behavior?
4. How does the presence of additional terms (lower-degree terms) affect the overall graph of the polynomial?
5. How does the multiplicity of roots influence the graph near the roots while maintaining the same end behavior?

### Tip:
When sketching the graph of a polynomial, always analyze the **degree**, **leading coefficient**, and **roots** to predict the overall shape and behavior.

Explore how the degree of a polynomial influences its end behavior. Learn about the role of leading coefficients, odd and even degrees, and their impact on the graph's behavior at extreme values. Suitable for high school students in Grades 10-12.

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