Question

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.
x
minus−2
0
2
4
​f(x)
1111
66
11
negative 4−4
Question content area bottom
Part 1
Select the correct choice​ and, if​ necessary, fill in the answer box in your choice below.
A.
The values could represent the linear function ​f(x)equals=enter your response here.
​(Use integers or fractions for any numbers in the​ expression.)

Answer 1

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.