Question

Which table represents a linear relationship
A. x f (x)
-1 -3
0 -2
1 1
2 6
3 13
B. -1 1/2
0 1
1 2
2 4
3 8
C. -1 -3
0 -1
1 1
2 3
3 5
D. -1 -1
0 0
1 1
2 8
3 27

Answer 1

To determine which table represents a linear relationship, we need to check if the change in the output value \( f(x) \) is consistent for equal changes in the input value \( x \). In a linear relationship, the slope (the change in \( f(x) \) divided by the change in \( x \)) should be constant.

Let's analyze each option:

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

Calculate changes:
- From -1 to 0: \( -2 + 3 = 1 \) (change = 1)
- From 0 to 1: \( 1 + 2 = 3 \) (change = 3)
- From 1 to 2: \( 6 - 1 = 5 \) (change = 5)
- From 2 to 3: \( 13 - 6 = 7 \) (change = 7)

The changes are not constant, so this is not linear.

---

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

Calculate changes:
- From -1 to 0: \( 1 - \frac{1}{2} = \frac{1}{2} \) (change = 0.5)
- From 0 to 1: \( 2 - 1 = 1 \) (change = 1)
- From 1 to 2: \( 4 - 2 = 2 \) (change = 2)
- From 2 to 3: \( 8 - 4 = 4 \) (change = 4)

The changes are not constant, so this is not linear.

---

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

Calculate changes:
- From -1 to 0: \( -1 + 3 = 2 \) (change = 2)
- From 0 to 1: \( 1 + 1 = 2 \) (change = 2)
- From 1 to 2: \( 3 - 1 = 2 \) (change = 2)
- From 2 to 3: \( 5 - 3 = 2 \) (change = 2)

All changes are constant \( (2) \), so this is linear.

---

**D.**
\[
\begin{array}{c|c}
x & f(x) \\
\hline
-1 & -1 \\
0 & 0 \\
1 & 1 \\
2 & 8 \\
3 & 27 \\
\end{array}
\]

Calculate changes:
- From -1 to 0: \( 0 + 1 = 1 \) (change = 1)
- From 0 to 1: \( 1 - 0 = 1 \) (change = 1)
- From 1 to 2: \( 8 - 1 = 7 \) (change = 7)
- From 2 to 3: \( 27 - 8 = 19 \) (change = 19)

The changes are not constant, so this is not linear.

---

Based on the analysis, the table that represents a linear relationship is:

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