Question

Identify the two tables which represent quadratic relationships.

x 0 1 2 3
y -2 0 2 4

x 0 1 2 3
y -2 -4 -8 -16

x 0 1 2 3
y 4 -4 -4 4

x 0 1 2 3
y 1 2 4 8

x 0 1 2 3
y -4 -8 -10 -10

x 0 1 2 3
y 3 4 5 6

Answer 1

To identify which tables represent quadratic relationships, we can look for a constant second difference in the \(y\) values as \(x\) increases.

Let's check each table step by step:

1. **First Table**:
- \( y = -2, 0, 2, 4 \)
- First differences: \(0 - (-2) = 2\), \(2 - 0 = 2\), \(4 - 2 = 2\)
- Second differences: \(2 - 2 = 0\), \(2 - 2 = 0\)
- **Constant second difference. This is a quadratic relationship.**

2. **Second Table**:
- \( y = -2, -4, -8, -16 \)
- First differences: \(-4 - (-2) = -2\), \(-8 - (-4) = -4\), \(-16 - (-8) = -8\)
- Second differences: \(-4 - (-2) = -2\), \(-8 - (-4) = -4\)
- **Not a quadratic relationship.**

3. **Third Table**:
- \( y = 4, -4, -4, 4 \)
- First differences: \(-4 - 4 = -8\), \(-4 - (-4) = 0\), \(4 - (-4) = 8\)
- Second differences: \(0 - (-8) = 8\), \(8 - 0 = 8\)
- **Constant second difference. This is a quadratic relationship.**

4. **Fourth Table**:
- \( y = 1, 2, 4, 8 \)
- First differences: \(2 - 1 = 1\), \(4 - 2 = 2\), \(8 - 4 = 4\)
- Second differences: \(2 - 1 = 1\), \(4 - 2 = 2\)
- **Not a quadratic relationship.**

5. **Fifth Table**:
- \( y = -4, -8, -10, -10 \)
- First differences: \(-8 - (-4) = -4\), \(-10 - (-8) = -2\), \(-10 - (-10) = 0\)
- Second differences: \(-2 - (-4) = 2\), \(0 - (-2) = 2\)
- **Not a quadratic relationship.**

6. **Sixth Table**:
- \( y = 3, 4, 5, 6 \)
- First differences: \(4 - 3 = 1\), \(5 - 4 = 1\), \(6 - 5 = 1\)
- Second differences: \(1 - 1 = 0\), \(1 - 1 = 0\)
- **Not a quadratic relationship.**

**Conclusion**: The two tables that represent quadratic relationships are the **first table** and **the third table**.