Question

is the table Linear, Quadratic, Exponential?

x y
1 0
2 -1
3 0
4 3
5 8

Answer 1

To determine whether the relationship between \(x\) and \(y\) in the table is linear, quadratic, or exponential, we can analyze the values:

```
x | y
---------
1 | 0
2 | -1
3 | 0
4 | 3
5 | 8
```

**Step 1: Check for a Linear Relationship**

For a linear relationship, the change in \(y\) should be constant for a constant change in \(x\). We can calculate the differences:

- Between \(x = 1\) and \(x = 2\): \(y\) changes from 0 to -1 (change of -1)
- Between \(x = 2\) and \(x = 3\): \(y\) changes from -1 to 0 (change of +1)
- Between \(x = 3\) and \(x = 4\): \(y\) changes from 0 to 3 (change of +3)
- Between \(x = 4\) and \(x = 5\): \(y\) changes from 3 to 8 (change of +5)

The differences are not constant; hence, it is not linear.

**Step 2: Check for a Quadratic Relationship**

For a quadratic relationship, we can check the second differences:

First differences (change in \(y\)):
- From 0 to -1 = -1
- From -1 to 0 = +1
- From 0 to 3 = +3
- From 3 to 8 = +5

First differences: \(-1, +1, +3, +5\)

Now, let's calculate the second differences:
- From -1 to +1 = +2
- From +1 to +3 = +2
- From +3 to +5 = +2

The second differences are constant (+2), which indicates that the relationship is quadratic.

**Step 3: Check for an Exponential Relationship**

For an exponential relationship, the ratio of consecutive \(y\) values should be constant, or the values should follow a pattern of \(y = ab^x\).

Calculate the ratios:
- From \(y = 0\) to \(y = -1\): Not applicable since \(y\) starts from 0.
- From \(y = -1\) to \(y = 0\): Not applicable.
- It is clear that these values do not form a consistent multiplication pattern, so it is not exponential.

**Conclusion:**

The relationship is **quadratic**.