Question

Which of the following best models the data in the table?

Type of Function

The table presents values of x and y, showing how y increases as x increases
x y
2
8.5

4
46

6
107.5

8
193

(1 point)
Responses

quadratic
quadratic

linear
linear

exponential
exponential

cubic
cubic
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Answer 1

To determine the best model that fits the given data points, let's look at the values:

```
x | y
------------
2 | 8.5
4 | 46
6 | 107.5
8 | 193
```

1. **Linear Function**: A linear function has the form \( y = mx + b \), which would indicate a constant rate of change. By checking the differences between y-values, we see they are not consistent, indicating it's not linear.

2. **Quadratic Function**: A quadratic function is of the form \( y = ax^2 + bx + c \). We can examine the second differences of y-values. If the second differences are constant, it would indicate a quadratic behavior.

3. **Exponential Function**: An exponential function might have the form \( y = ab^x \), which would imply that the rate of change increases rapidly.

4. **Cubic Function**: A cubic function is of the form \( y = ax^3 + bx^2 + cx + d \) and would show a more complex relation with varying rate of change.

Let's check the first and second differences in y-values:

- **First differences** (subtract consecutive y-values):

- \( 46 - 8.5 = 37.5 \)
- \( 107.5 - 46 = 61.5 \)
- \( 193 - 107.5 = 85.5 \)

- The differences: \( 37.5, 61.5, 85.5 \)

- **Second differences** (subtract consecutive first differences):

- \( 61.5 - 37.5 = 24 \)
- \( 85.5 - 61.5 = 24 \)

The second differences are constant, suggesting that the relationship between x and y can be modeled by a quadratic function.

Based on the analysis, the best fit for the data is:

**Quadratic Function**.