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

cubic
cubic

linear
linear

exponential
exponential

quadratic

Answer 1

To determine which type of function best models the data, we can analyze the rate of change in the values of \(y\) as \(x\) increases.

Given the pairs from the table:

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

1. **Linear Function:** A linear function would show a constant rate of change. However, we can calculate the differences:

- From \(y=8.5\) to \(y=46\): \(46 - 8.5 = 37.5\)
- From \(y=46\) to \(y=107.5\): \(107.5 - 46 = 61.5\)
- From \(y=107.5\) to \(y=193\): \(193 - 107.5 = 85.5\)

The differences are not constant, ruling out a linear function.

2. **Quadratic Function:** A quadratic function would have a second difference that is constant. Here are the first differences again:

- First differences: \(37.5\), \(61.5\), \(85.5\)
- Second differences:
- \(61.5 - 37.5 = 24\)
- \(85.5 - 61.5 = 24\)

The second difference is consistent, suggesting that a quadratic function might be appropriate.

3. **Cubic Function and Exponential Function:**
- A cubic function would generally show a changing second difference. The second difference is consistent, so a cubic model is not suggested here.
- An exponential function tends to grow at an increasing rate, but the values do not appear to follow an exponential growth pattern since the differences grow but in a way that aligns more with a quadratic trend.

Based on this analysis, **quadratic** is the most suitable type of function to model this data. Therefore, the answer is:

**quadratic**