Question

Use differences or ratios to tell whether the table of values represents a linear function, an exponential function, or a quadratic function.
x|-2|-1|0|1|2
y|-6|-6|-4|0|6
First differences:
Second differences:

Answer 1

To determine whether the given table of values represents a linear function, an exponential function, or a quadratic function, we can calculate the first differences and second differences of the \( y \) values.

### Given Table:

| x | y |
|-----|-----|
| -2 | -6 |
| -1 | -6 |
| 0 | -4 |
| 1 | 0 |
| 2 | 6 |

### Calculate First Differences:

The first difference is calculated by subtracting consecutive \( y \) values:

- First difference between \( y(-1) \) and \( y(-2) \): \(-6 - (-6) = 0\)
- First difference between \( y(0) \) and \( y(-1) \): \(-4 - (-6) = 2\)
- First difference between \( y(1) \) and \( y(0) \): \(0 - (-4) = 4\)
- First difference between \( y(2) \) and \( y(1) \): \(6 - 0 = 6\)

First differences: \( 0, 2, 4, 6 \)

### Calculate Second Differences:

The second difference is calculated by subtracting consecutive first differences:

- Second difference between \( 2 \) and \( 0 \): \(2 - 0 = 2\)
- Second difference between \( 4 \) and \( 2 \): \(4 - 2 = 2\)
- Second difference between \( 6 \) and \( 4 \): \(6 - 4 = 2\)

Second differences: \( 2, 2, 2 \)

### Conclusion:

- The first differences are not constant, meaning it is not a linear function.
- The second differences are constant, which means the values represent a **quadratic function**.

So, the table of values represents a **quadratic function**.