Question

Is the table linear, quadratic, or exponential?

A.
x|1|2|3|4|5
y|5|9|13|17|21

B.
x|1|2|3|4|5
y|0|-1|0|3|8

C.
x|1|2|3|4|5
y|3|9|27|81|243

Answer 1

To determine whether the relations in the tables are linear, quadratic, or exponential, we can analyze the patterns in the given data.

**A.**
```
x | 1 | 2 | 3 | 4 | 5
y | 5 | 9 | 13 | 17 | 21
```
To check if it's linear, we can find the differences in \(y\):
- \(9 - 5 = 4\)
- \(13 - 9 = 4\)
- \(17 - 13 = 4\)
- \(21 - 17 = 4\)

The first differences are constant (\(4\)), indicating that this relation is **linear**.

---

**B.**
```
x | 1 | 2 | 3 | 4 | 5
y | 0 | -1 | 0 | 3 | 8
```
Next, we'll find the differences in \(y\):
- \(-1 - 0 = -1\)
- \(0 - (-1) = 1\)
- \(3 - 0 = 3\)
- \(8 - 3 = 5\)

The first differences are: \(-1, 1, 3, 5\).

Now checking the second differences:
- \(1 - (-1) = 2\)
- \(3 - 1 = 2\)
- \(5 - 3 = 2\)

The second differences are constant, indicating that this relation is **quadratic**.

---

**C.**
```
x | 1 | 2 | 3 | 4 | 5
y | 3 | 9 | 27 | 81 | 243
```
For this, we can check the ratio of \(y\) values:
- \(9 / 3 = 3\)
- \(27 / 9 = 3\)
- \(81 / 27 = 3\)
- \(243 / 81 = 3\)

The ratios are constant (all \(3\)), indicating that this relation is **exponential**.

---

**Summary:**
- A is linear.
- B is quadratic.
- C is exponential.