Use the table below to determine whether​ f(x) could represent a linear function. If it​ could, write​ f(x) in the form ​f(x)equals=mxplus+b.

To determine if the function \( f(x) \) could be linear, we need to check if the differences in \( f(x) \) values are consistent for equal differences in \( x \).

Let's fill in the provided data into a table:

\[ \begin{array}{|c|c|} \hline x & f(x) \ \hline -2 & 11 \ 0 & 6 \ 2 & 11 \ 4 & -4 \ \hline \end{array} \]

Step 1: Calculate the slopes between the points.

The slope between two points \( (x_1, f(x_1)) \) and \( (x_2, f(x_2)) \) is given by the formula:

\[ m = \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]

Calculate slopes between consecutive points:

1. Between \( (-2, 11) \) and \( (0, 6) \): \[ m = \frac{6 - 11}{0 - (-2)} = \frac{-5}{2} = -\frac{5}{2} \]

2. Between \( (0, 6) \) and \( (2, 11) \): \[ m = \frac{11 - 6}{2 - 0} = \frac{5}{2} \]

3. Between \( (2, 11) \) and \( (4, -4) \): \[ m = \frac{-4 - 11}{4 - 2} = \frac{-15}{2} \]

Step 2: Analyze the slopes.

The slopes calculated are:

• Between \( (-2, 11) \) and \( (0, 6) \): \( -\frac{5}{2} \)
• Between \( (0, 6) \) and \( (2, 11) \): \( \frac{5}{2} \)
• Between \( (2, 11) \) and \( (4, -4) \): \( -\frac{15}{2} \)

Since the slopes are not consistent (they differ), \( f(x) \) does not represent a linear function.

Part 1 Answer:

A. The values could not represent the linear function \( f(x) \).

If we fill in a response, the final answer would be: A. The values could not represent a linear function.