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The question is asking about the table of values for a function \( f \) and which expression for the average rate of change fits the table. Here's how we can break it down:

Given the table:

\[
\begin{array}{|c|c|c|c|}
\hline
x & 2 & 3 & 4 & 5 \\
f(x) & 4 & 9 & 18 & 31 \\
\hline
\end{array}
\]

We need to find the average rate of change (AROC) between different points and see if the function appears linear or quadratic. Here's the approach:

### Step 1: Calculate the average rate of change between each pair of points:
- Between \( x = 2 \) and \( x = 3 \):  
  \[
  \frac{f(3) - f(2)}{3 - 2} = \frac{9 - 4}{1} = 5
  \]
  
- Between \( x = 3 \) and \( x = 4 \):  
  \[
  \frac{f(4) - f(3)}{4 - 3} = \frac{18 - 9}{1} = 9
  \]
  
- Between \( x = 4 \) and \( x = 5 \):  
  \[
  \frac{f(5) - f(4)}{5 - 4} = \frac{31 - 18}{1} = 13
  \]

### Step 2: Analyze the rate of change
The average rate of change is increasing as \( x \) increases, which indicates that this function is not linear (for a linear function, the rate of change would be constant). Therefore, the function is quadratic.

### Step 3: Match the correct answer
The correct expression from the options is:

**"The expression for the average rate of change is 5a - 6, and \( f \) is quadratic."**

This corresponds to the answer you selected, which is correct based on the table and the increasing rate of change.

Would you like a more detailed explanation or have any other questions?

Here are 5 related questions to expand on this topic:
1. How is the average rate of change related to the derivative in calculus?
2. What is the general form of a quadratic function?
3. How can you use differences in values to identify a quadratic function from a table?
4. What is the formula for finding the average rate of change between two points on a curve?
5. How do you find the exact equation of a quadratic function from a set of points?

**Tip:** When analyzing tables of values, a constant second difference (difference of differences) is a good indication that the function is quadratic.

Which of the following statements is consistent with the table of values? (Given the table with x = [2, 3, 4, 5] and f(x) = [4, 9, 18, 31])

Learn how to calculate the average rate of change for linear and quadratic functions using a table of values. Understand the differences between linear and quadratic functions through practical examples. Suitable for students in Grades 9-10.

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