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please help! A and C are wrong! please i need a 100%

Let's analyze the data and solve it step by step.

The table gives intervals and their corresponding Average Rate of Change (AROC):

- Interval: \([-2, 0], [0, 2], [2, 4], [4, 6], [6, 8], [8, 10]\)
- AROC: \([20, 12, 100, 284, 564, 940]\)

### Approach:

To determine the appropriate model (linear, quadratic, cubic, or quartic), we need to evaluate how the AROC changes with respect to each interval:

1. **Linear model** would imply the first differences of AROC are constant (Δy/Δx is constant).
2. **Quadratic model** implies the second differences are constant (Δ²y/Δx² is constant).
3. **Cubic model** implies the third differences are constant (Δ³y/Δx³ is constant).
4. **Quartic model** implies the fourth differences are constant (Δ⁴y/Δx⁴ is constant).

### Step-by-step differences:

- **First differences (Δy)**:
  \[
  12 - 20 = -8, \quad 100 - 12 = 88, \quad 284 - 100 = 184, \quad 564 - 284 = 280, \quad 940 - 564 = 376
  \]
  So the first differences are: \([-8, 88, 184, 280, 376]\)

- **Second differences (Δ²y)**:
  \[
  88 - (-8) = 96, \quad 184 - 88 = 96, \quad 280 - 184 = 96, \quad 376 - 280 = 96
  \]
  The second differences are all constant: \([96, 96, 96, 96]\).

Since the **second differences are constant**, the function is quadratic. Therefore, the correct model is the **quadratic model**.

The answer is:

- The quadratic model is appropriate because \(\frac{Δ²y}{Δx²}\) is constant.

Would you like more details on this process, or any further clarifications?

### Related questions:
1. What are the differences between a quadratic and a cubic model?
2. How can we determine if a model fits the data perfectly?
3. How does the AROC relate to the derivative of a function?
4. What role do second-order differences play in fitting polynomial models?
5. How can this technique be applied to other types of functions?

### Tip:
When analyzing AROC tables, always start by computing first and second differences to determine if the model is quadratic, as quadratic models often appear in real-world situations!

This problem helps determine the appropriate polynomial model based on the average rate of change (AROC) for selected intervals. The second differences are constant, indicating the quadratic model is most suitable. Suitable for Grades 10-12.

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