Question

Decide whether the data in the table represents a linear function or an exponential function. Explain how you know.
X | Y
1 | 4
2 | -5
3 | -14
4 | -23
5 | -32
This table represents a(n) ________ function because there is a common __________ of ____

Answer 1

To determine whether the data in the table represents a linear function or an exponential function, we can analyze how the values of \(Y\) change as \(X\) increases.

### Given Data:
```
X | Y
1 | 4
2 | -5
3 | -14
4 | -23
5 | -32
```

### Steps to Analyze:

1. **Calculate the differences in \(Y\):**
- From \(Y(1)\) to \(Y(2)\): \(-5 - 4 = -9\)
- From \(Y(2)\) to \(Y(3)\): \(-14 - (-5) = -9\)
- From \(Y(3)\) to \(Y(4)\): \(-23 - (-14) = -9\)
- From \(Y(4)\) to \(Y(5)\): \(-32 - (-23) = -9\)

The differences in \(Y\) when \(X\) increases by 1 are constant, equal to \(-9\).

2. **Conclusion:**
Since the change in \(Y\) with each increase in \(X\) is constant, the data represents a linear function.

### Final Statement:
This table represents a **linear** function because there is a common **difference** of **-9**.