Let's go through each part of the solution step-by-step.
### Part (a): Finding a Power Function that Models the Data
Given the data:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 5 \\
\hline
2 & 8 \\
\hline
3 & 13 \\
\hline
4 & 21 \\
\hline
5 & 32 \\
\hline
6 & 45 \\
\hline
\end{array}
\][/tex]
We are looking for a power function of the form [tex]\( y = a \cdot x^b \)[/tex].
The parameters for the power function are found to be [tex]\( a \approx 2.116 \)[/tex] and [tex]\( b \approx 1.696 \)[/tex].
So, the power function that models the data is:
[tex]\( y = 2.116 \cdot x^{1.696} \)[/tex].
### Part (b): Finding a Linear Function that Models the Data
We need a linear function of the form [tex]\( y = mx + c \)[/tex].
The parameters for the linear function are found to be [tex]\( m \approx 8.000 \)[/tex] and [tex]\( c \approx -7.333 \)[/tex].
Thus, the linear function that models the data is:
[tex]\( y = 8.000 \cdot x - 7.333 \)[/tex].
### Part (c): Determining Which Function is the Better Fit
To determine which function better fits the data, we'll compare them visually or using an error metric like Residual Sum of Squares (RSS). The function with the lower RSS is typically considered a better fit.
After comparing the two functions:
- The power function parameters are:
[tex]\( a \approx 2.116 \)[/tex] and [tex]\( b \approx 1.696 \)[/tex], leading to the function [tex]\( y = 2.116 \cdot x^{1.696} \)[/tex].
- The linear function parameters are:
[tex]\( m \approx 8.000 \)[/tex] and [tex]\( c \approx -7.333 \)[/tex], leading to the function [tex]\( y = 8.000 \cdot x - 7.333 \)[/tex].
Visually inspecting the data and the resulting functions (or using an error metric), we conclude that the power function provides the better fit for this specific data set compared to the linear function.
Therefore, summary of our results is:
a. The power function is:
[tex]\[
y = 2.116 \cdot x^{1.696}
\][/tex]
b. The linear function is:
[tex]\[
y = 8.000 \cdot x - 7.333
\][/tex]
c. The power function is determined to be the better fit for the data.
