To determine the type of function that best models the given data, we should analyze how the values of y change as x increases.
A linear function takes the form \( y = mx + b \) where the rate of change (slope) is constant.
A quadratic function has the form \( y = ax^2 + bx + c \), where the rate of change is not constant, but the second differences are constant.
A cubic function has the form \( y = ax^3 + bx^2 + cx + d \), which leads to an increasingly changing rate of increase.
An exponential function has the form \( y = a \cdot b^x \), where y increases at an increasing rate.
Let's look at the given values:
- At \( x = 2 \), \( y = 8.5 \)
- At \( x = 4 \), \( y = 46 \)
- At \( x = 6 \), \( y = 107.5 \)
- At \( x = 8 \), \( y = 193 \)
To check if the growth is linear, we can observe the differences between consecutive y values:
- 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 differences (37.5, 61.5, 85.5) are not constant, indicating that this is not a linear function.
Next, let's check if the growth could be quadratic by looking at the second differences:
- First differences: \( 37.5, 61.5, 85.5 \)
- 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), which suggests that the data might be well represented by a quadratic function.
In conclusion, based on the analysis of the differences, the best model for the data is quadratic.