Question

is the table Linear, Quadratic, Exponential?

x y
1 3
2 9
3 27
4 81
5 243

Answer 1

To determine the type of relationship represented by the table of values, we can analyze the pattern in the \( y \) values as \( x \) increases:

1. **Check for a Linear Relationship:**
A linear relationship between \( x \) and \( y \) would mean that the difference between successive \( y \) values is constant.
- \( y(2) - y(1) = 9 - 3 = 6 \)
- \( y(3) - y(2) = 27 - 9 = 18 \)
- \( y(4) - y(3) = 81 - 27 = 54 \)
- \( y(5) - y(4) = 243 - 81 = 162 \)

The differences (6, 18, 54, 162) are not constant, so the relationship is not linear.

2. **Check for a Quadratic Relationship:**
A quadratic relationship would mean that the second difference of the \( y \) values is constant.
- First differences: 6, 18, 54, 162
- Second differences:
- \( 18 - 6 = 12 \)
- \( 54 - 18 = 36 \)
- \( 162 - 54 = 108 \)

The second differences (12, 36, 108) are not constant, so the relationship is also not quadratic.

3. **Check for an Exponential Relationship:**
An exponential relationship would exhibit a constant ratio between successive \( y \) values.
- Ratios:
- \( \frac{y(2)}{y(1)} = \frac{9}{3} = 3 \)
- \( \frac{y(3)}{y(2)} = \frac{27}{9} = 3 \)
- \( \frac{y(4)}{y(3)} = \frac{81}{27} = 3 \)
- \( \frac{y(5)}{y(4)} = \frac{243}{81} = 3 \)

The ratios (3, 3, 3, 3) are all constant, which indicates an exponential relationship.

**Conclusion:** The relationship represented by the table is **exponential**. Specifically, the pattern follows \( y = 3^x \).