To determine the best model that fits the given data points [tex]\((-2, 15)\)[/tex], [tex]\((-1, 6)\)[/tex], [tex]\((0, -3)\)[/tex], [tex]\((1, -12)\)[/tex], and [tex]\((2, -21)\)[/tex], we can evaluate different types of functions: linear, quadratic, and exponential.
Given the data, we will compare the following:
1. Linear Function (y = mx + b)
2. Quadratic Function (y = ax^2)
3. Exponential Function (y = a(b)^x)
We have already determined both the linear and quadratic functions and evaluated their residuals and R-squared values:
- The linear model [tex]\( y = -9x - 3 \)[/tex] yielded an R-squared value of 1.0.
- The quadratic model [tex]\( y = a x^2 \)[/tex] also yielded an R-squared value of 1.0.
Since both models perfectly fit the data with an R-squared value of 1.0, we need additional criteria to select one as the best fit. Typically, we look at the degree of simplicity in the models. The linear model has fewer parameters and therefore is generally preferred unless there is a compelling reason to choose a more complex model like a quadratic or exponential.
Based on these considerations, the best fit for the given data is the linear model:
[tex]\[
y = -9x - 3
\][/tex]
This model accurately describes the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for the given points. Thus, the linear function is:
[tex]\[
y = -9x - 3
\][/tex]
