To find the exponential function that best models the given data, we need to fit an exponential curve to the points provided. The data points are:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Time (years)} & \text{Account Value (\$)} \\
\hline
0 & 400 \\
\hline
1 & 436 \\
\hline
2 & 468 \\
\hline
3 & 502 \\
\hline
4 & 550 \\
\hline
5 & 589 \\
\hline
\end{array}
\][/tex]
To model this data with an exponential function, we use the general form of the exponential function:
[tex]\[ f(x) = a (b^x) \][/tex]
where [tex]\( a \)[/tex] is the initial value (the account value at [tex]\( x = 0 \)[/tex]) and [tex]\( b \)[/tex] is the growth factor.
1. Determine the initial value [tex]\( a \)[/tex]
From the table, we see that at [tex]\( x = 0 \)[/tex], the account value is [tex]\(\$400\)[/tex]. So, [tex]\( a = 400 \)[/tex].
2. Determine the growth factor [tex]\( b \)[/tex]
We observe the changes in the account value over time. The exponential growth factor, [tex]\( b \)[/tex], can be estimated by fitting the exponential curve to the given data points. By fitting an exponential curve, we find the parameter [tex]\( b \)[/tex] that best describes the growth of the account value over the different time points.
After analyzing the given data:
- At [tex]\( x = 0 \)[/tex], value is \[tex]$400.
- At \( x = 1 \), value is \$[/tex]436.
- At [tex]\( x = 2 \)[/tex], value is \[tex]$468.
- At \( x = 3 \), value is \$[/tex]502.
- At [tex]\( x = 4 \)[/tex], value is \[tex]$550.
- At \( x = 5 \), value is \$[/tex]589.
4. Identify the exponential function from the given choices:
Upon analyzing various fits, we found that the closest match for the parameters [tex]\( a \)[/tex] and [tex]\( b \)[/tex] (when rounded to the nearest hundredth) are:
[tex]\[ a \approx 401.19 \][/tex]
[tex]\[ b \approx 1.08 \][/tex]
Thus, the exponential function that models the data is:
[tex]\[ f(x) = 401.19 (1.08^x) \][/tex]
Hence, the correct option is:
[tex]\[ f(x) = 401.19(1.08)^x \][/tex]
