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

Question

Type of Function

x y
2
8.5

4
46

6
107.5

8
193
(1 point)
Responses

quadratic
quadratic

linear
linear

exponential
exponential

cubic

Answer 1

To determine the best model for the given data, we need to analyze the relationship between \(x\) and \(y\) values.

Let's first list the provided data:

• When \(x = 2\), \(y = 8.5\)
• When \(x = 4\), \(y = 46\)
• When \(x = 6\), \(y = 107.5\)
• When \(x = 8\), \(y = 193\)

Step 1: Examine the differences

We can compute the differences in \(y\) values for subsequent \(x\) values:

1. \(y(4) - y(2) = 46 - 8.5 = 37.5\)
2. \(y(6) - y(4) = 107.5 - 46 = 61.5\)
3. \(y(8) - y(6) = 193 - 107.5 = 85.5\)

Step 2: Examine the second differences

Now, let’s compute the differences of the differences:

1. \(61.5 - 37.5 = 24\)
2. \(85.5 - 61.5 = 24\)

The second differences are constant (24), which suggests that the function is quadratic.

Conclusion

Since the second differences are constant, the best model for the data in the table is a quadratic function.

Thus, the answer is:

Quadratic