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Let's analyze the functions and their end behavior to place them in the correct row of the table.

### Key Concepts:
- The **end behavior** of a polynomial function is determined by the **degree** of the function (highest exponent of \( x \)) and the **sign of the leading coefficient** (the coefficient of the term with the highest degree).
  
- **For even degree functions:**
  - If the leading coefficient is positive, the graph extends from quadrant 2 to quadrant 1.
  - If the leading coefficient is negative, the graph extends from quadrant 3 to quadrant 4.
  
- **For odd degree functions:**
  - If the leading coefficient is positive, the graph extends from quadrant 3 to quadrant 1.
  - If the leading coefficient is negative, the graph extends from quadrant 2 to quadrant 4.

### Functions Given:
1. \( y = 2x \) (Degree 1, positive leading coefficient)
2. \( y = 5x^6 \) (Degree 6, positive leading coefficient)
3. \( y = -3x^2 \) (Degree 2, negative leading coefficient)
4. \( y = x^7 \) (Degree 7, positive leading coefficient)
5. \( y = -\frac{2}{5}x^9 \) (Degree 9, negative leading coefficient)
6. \( y = -4x^5 \) (Degree 5, negative leading coefficient)
7. \( y = x^{10} \) (Degree 10, positive leading coefficient)
8. \( y = -0.5x^8 \) (Degree 8, negative leading coefficient)

### Filling the Table:
#### Row 1: Extends from quadrant 3 to quadrant 1
- **Function:** \( y = x^7 \), \( y = 2x \)
- **Reason:** Both have odd degrees and positive leading coefficients.

#### Row 2: Extends from quadrant 3 to quadrant 4
- **Function:** \( y = -4x^5 \), \( y = -\frac{2}{5}x^9 \)
- **Reason:** Both have odd degrees and negative leading coefficients.

#### Row 3: Extends from quadrant 2 to quadrant 1
- **Function:** \( y = 5x^6 \), \( y = x^{10} \)
- **Reason:** Both have even degrees and positive leading coefficients.

#### Row 4: Extends from quadrant 3 to quadrant 4
- **Function:** \( y = -3x^2 \), \( y = -0.5x^8 \)
- **Reason:** Both have even degrees and negative leading coefficients.

### Completed Table:

| End Behaviour                               | Function                         | Reasons                                               |
|---------------------------------------------|----------------------------------|-------------------------------------------------------|
| Extends from quadrant 3 to quadrant 1       | \( y = x^7 \), \( y = 2x \)      | Odd degree, positive leading coefficient               |
| Extends from quadrant 3 to quadrant 4       | \( y = -4x^5 \), \( y = -\frac{2}{5}x^9 \) | Odd degree, negative leading coefficient               |
| Extends from quadrant 2 to quadrant 1       | \( y = 5x^6 \), \( y = x^{10} \) | Even degree, positive leading coefficient              |
| Extends from quadrant 3 to quadrant 4       | \( y = -3x^2 \), \( y = -0.5x^8 \) | Even degree, negative leading coefficient              |

Would you like further explanations on any of the concepts or functions involved?

### Five Related Questions:
1. What happens to the end behavior of a polynomial when both degree and leading coefficient are negative?
2. How do the degrees of polynomials affect their graph shapes and turning points?
3. Can you predict the number of real roots based on the degree of a polynomial?
4. How do higher-degree polynomials differ in their end behaviors compared to lower-degree ones?
5. What is the significance of the leading term in determining the graph's end behavior?

**Tip:** Always check both the degree and sign of the leading term to determine the end behavior of any polynomial.

Write each function in the appropriate row of the second column of the table. Give reasons for your choices.

Learn how to classify polynomial functions based on their end behavior by analyzing their degree and the sign of their leading coefficient. This example includes functions like y = 2x, y = 5x^6, and y = x^7, making it suitable for students in Grades 10-12.

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