Question

Which table of values represents a linear relationship
x f(x) x f(x) x f (x) x f(x)
-1 -3 -1 1/2 -1 -3 -1 -1
0 -2 0 1 0 -1 0 0
1 1 1 2 1 1 1 1
2 6 2 4 2 3 2 8
3 13 3 8 3 5 3 27

Answer 1

To determine which table of values represents a linear relationship, we can check if the differences between consecutive \( f(x) \) values are constant for each table. In a linear relationship, the change in \( f(x) \) (the output) is proportional to the change in \( x \) (the input).

Let's analyze each table:

### Table 1
\[
\begin{array}{c|c}
x & f(x) \\
\hline
-1 & -3 \\
0 & -2 \\
1 & 1 \\
2 & 6 \\
3 & 13 \\
\end{array}
\]

Calculate differences:
- From -3 to -2: \( -2 - (-3) = 1 \)
- From -2 to 1: \( 1 - (-2) = 3 \)
- From 1 to 6: \( 6 - 1 = 5 \)
- From 6 to 13: \( 13 - 6 = 7 \)

The differences are not constant (1, 3, 5, 7), so **this table does not represent a linear relationship.**

### Table 2
\[
\begin{array}{c|c}
x & f(x) \\
\hline
-1 & \frac{1}{2} \\
0 & 1 \\
1 & 2 \\
2 & 4 \\
3 & 8 \\
\end{array}
\]

Calculate differences:
- From \( \frac{1}{2} \) to 1: \( 1 - \frac{1}{2} = \frac{1}{2} \)
- From 1 to 2: \( 2 - 1 = 1 \)
- From 2 to 4: \( 4 - 2 = 2 \)
- From 4 to 8: \( 8 - 4 = 4 \)

The differences are not constant (\( \frac{1}{2}, 1, 2, 4 \)), so **this table does not represent a linear relationship.**

### Table 3
\[
\begin{array}{c|c}
x & f(x) \\
\hline
-1 & -3 \\
0 & -1 \\
1 & 1 \\
2 & 3 \\
3 & 5 \\
\end{array}
\]

Calculate differences:
- From -3 to -1: \( -1 - (-3) = 2 \)
- From -1 to 1: \( 1 - (-1) = 2 \)
- From 1 to 3: \( 3 - 1 = 2 \)
- From 3 to 5: \( 5 - 3 = 2 \)

The differences are constant (2), so **this table represents a linear relationship.**

### Table 4
\[
\begin{array}{c|c}
x & f(x) \\
\hline
-1 & -1 \\
0 & 0 \\
1 & 1 \\
2 & 8 \\
3 & 27 \\
\end{array}
\]

Calculate differences:
- From -1 to 0: \( 0 - (-1) = 1 \)
- From 0 to 1: \( 1 - 0 = 1 \)
- From 1 to 8: \( 8 - 1 = 7 \)
- From 8 to 27: \( 27 - 8 = 19 \)

The differences are not constant (1, 1, 7, 19), so **this table does not represent a linear relationship.**

### Conclusion
The only table that represents a linear relationship is **Table 3**.