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

cubic
cubic

exponential
exponential

linear

Answer 1

To determine which type of function best models the data, we need to observe how the values of \(y\) change as \(x\) increases. Let's analyze the differences between the \(y\) values:

1. **Calculate the differences in \(y\):**
- From \(8.5\) to \(46\): \(46 - 8.5 = 37.5\)
- From \(46\) to \(107.5\): \(107.5 - 46 = 61.5\)
- From \(107.5\) to \(193\): \(193 - 107.5 = 85.5\)

The first differences are: \(37.5, 61.5, 85.5\).

2. **Calculate the second differences:**
- From \(37.5\) to \(61.5\): \(61.5 - 37.5 = 24\)
- From \(61.5\) to \(85.5\): \(85.5 - 61.5 = 24\)

The second differences are constant: \(24\).

Since the second differences are constant, this indicates that the relationship between \(x\) and \(y\) is best modeled by a quadratic function.

Thus, the best answer from the options provided is:

**quadratic**.