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The question presents a table with the **average rate of change (AROC)** over different intervals of a polynomial function, and asks which model is most appropriate for the data, suggesting four possible options: **linear, quadratic, cubic, or quartic** models. Let's break this down:

### Given Data:
- **Intervals**: \([-2, 0], [0, 2], [2, 4], [4, 6], [6, 8], [8, 10]\)
- **AROC values**: 20, 12, 100, 284, 564, 940

The key idea is to find which derivative (first, second, third, or fourth) yields a constant value. For each type of polynomial:

1. **Linear**: The **first difference** (Δy/Δx) should be constant.
2. **Quadratic**: The **second difference** should be constant.
3. **Cubic**: The **third difference** should be constant.
4. **Quartic**: The **fourth difference** should be constant.

### Step-by-Step Differences:
1. **First difference** (Δy):  
   (12 - 20), (100 - 12), (284 - 100), (564 - 284), (940 - 564)
   = -8, 88, 184, 280, 376

   This shows that the **first difference is not constant**, so a linear model is ruled out.

2. **Second difference** (Δ²y):  
   (88 - (-8)), (184 - 88), (280 - 184), (376 - 280)
   = 96, 96, 96, 96

   The **second difference is constant**, which suggests that the polynomial function is **quadratic**. 

### Conclusion:
The appropriate model for the data is **quadratic** because the second difference is constant. Thus, the correct answer is:

- "The quadratic model is appropriate because \(\Delta^2 y / \Delta x\) is constant."

Would you like further details or clarification on the process?

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Here are five related questions to expand understanding:
1. What is the significance of the second difference in a polynomial model?
2. How would the data change if the cubic model were correct?
3. Can you explain how to calculate the first and second differences in a table of data?
4. How do average rates of change relate to polynomial degrees?
5. What are real-world examples of quadratic models based on AROC?

**Tip:** For identifying polynomial degree, always start with the first differences and work your way up!

The table gives the average rate of change for selected intervals of a polynomial function. Which model is most appropriate for the data?

Learn how to determine the correct polynomial model based on the average rate of change over given intervals. This example uses differences to select between linear, quadratic, cubic, and quartic models.

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