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Let's break down and solve the questions from the table and the instructions given.

### Part A: Determine if function \( f \) is linear or quadratic according to the data in the table.
The table provides the average rate of change over three consecutive intervals:

| Interval  | Average Rate of Change |
|-----------|------------------------|
| \([-1, 1]\) | 2                      |
| \([1, 3]\)  | -2                     |
| \([3, 5]\)  | -6                     |

A **linear function** has a constant rate of change, but here the average rate of change is different for each interval. This suggests that the function is not linear.

A **quadratic function** has a changing rate of change, but the rate of change itself changes in a linear manner. We can check the differences between the average rates of change:

- Change from 2 to -2: \( -2 - 2 = -4 \)
- Change from -2 to -6: \( -6 - (-2) = -4 \)

Since the differences between the average rates of change are constant (both are \(-4\)), the function is likely quadratic.

### Part B:
**(i) On which interval, if any, is function \( f \) increasing?**

The function is increasing where the rate of change is positive. From the table, we see that the function is increasing on the interval \([-1, 1]\), where the average rate of change is 2.

**(ii) On which interval, if any, is function \( f \) decreasing?**

The function is decreasing where the rate of change is negative. From the table, the function is decreasing on the intervals \([1, 3]\) and \([3, 5]\), where the average rates of change are -2 and -6, respectively.

### Part C:
**(i) Give the rate of change or the expression of the rate of change for function \( f \) if it contains the ordered pair \( (1, -7) \). Show your work.**

We do not have sufficient data to directly compute the rate of change from this pair alone. However, if this ordered pair fits the quadratic model suggested earlier, we would need more context, such as a formula or additional points, to compute the specific rate of change.

**(ii) Is function \( f \) concave up, down, or neither on the interval \([-1, 5]\)?**

Since the rate of change is decreasing (the average rate of change goes from 2 to -6), the function is concave down on this interval. For a quadratic function, concave down means that the second derivative is negative.

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Do you want more detailed explanations for any part, or would you like assistance with specific aspects?

### 5 Follow-up Questions:
1. How do you determine if a function is quadratic from a table of rates of change?
2. Why is a constant second difference an indicator of a quadratic function?
3. What is the geometric interpretation of concavity for a function?
4. How does the rate of change relate to the behavior of a function on different intervals?
5. How would you find the exact equation of a quadratic function based on points and average rates of change?

### Tip:
Always check the second differences in rates of change when trying to determine if a function is quadratic!

The table gives the average rates of change of function f over consecutive equal-length intervals. Part A: Determine if function f is linear or quadratic according to the data in the table. Explain your reasoning. Part B: On which interval, if any, is function f increasing? On which interval, if any, is function f decreasing? Part C: Give the rate of change or the expression of the rate of change for function f if it contains the ordered pair (1, -7). Is function f concave up, down, or neither on the interval [-1, 5]? Explain your reasoning.

Learn how to determine whether a function is linear or quadratic by analyzing average rates of change over intervals. This problem covers key concepts such as concavity, increasing/decreasing intervals, and rate of change.

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