Looking at the given table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & 8 \\
\hline
2 & 16 \\
\hline
3 & 32 \\
\hline
4 & 64 \\
\hline
5 & 128 \\
\hline
\end{tabular}
\][/tex]
We can see a clear pattern: each value of \( y \) is double the previous value of \( y \). This indicates the relationship between \( x \) and \( y \) is exponential in nature, rather than linear.
An exponential function is typically of the form:
[tex]\[
y = a \cdot b^x
\][/tex]
To determine the values of the coefficients \( a \) and \( b \), we use the given data points.
From the table, we can see:
[tex]\[
y(1) = 8 \\
y(2) = 16 \\
y(3) = 32 \\
y(4) = 64 \\
y(5) = 128
\][/tex]
To find \( b \):
Taking the ratio of consecutive \( y \)-values to determine the base \( b \):
[tex]\[
\frac{y(2)}{y(1)} = \frac{16}{8} = 2
\][/tex]
Thus, \( b = 2 \).
To find \( a \):
Using the first data point \( (x = 1, y = 8) \):
[tex]\[
8 = a \cdot 2^1 \Rightarrow a = \frac{8}{2} = 4
\][/tex]
So, the exponential function that models the data is:
[tex]\[
\boxed{y = 4 \cdot 2^x}
\][/tex]
