To determine which exponential model best represents the given data, we need to measure how closely each model matches the data points. A standard method to achieve this is by calculating the sum of the squared errors (SSE) for each model. The model with the smallest SSE is considered the best fit.
Given the data points:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-2 & 128 \\
\hline
-1 & 27 \\
\hline
0 & 5 \\
\hline
1 & 1 \\
\hline
2 & 0.1 \\
\hline
\end{array}
\][/tex]
The candidate exponential models are:
1. [tex]\( f(x) = 5(1.2)^x \)[/tex]
2. [tex]\( f(x) = 5(0.2)^x \)[/tex]
3. [tex]\( f(x) = 2(5)^x \)[/tex]
4. [tex]\( f(x) = 2(0.5)^x \)[/tex]
Let's evaluate the sum of squared errors for each model by comparing the model's predicted values with the actual data points.
Model 1: [tex]\( f(x) = 5(1.2)^x \)[/tex]
Sum of squared errors (SSE): 16103.938549382714
Model 2: [tex]\( f(x) = 5(0.2)^x \)[/tex]
Sum of squared errors (SSE): 13.010000000000085
Model 3: [tex]\( f(x) = 2(5)^x \)[/tex]
Sum of squared errors (SSE): 19651.096400000002
Model 4: [tex]\( f(x) = 2(0.5)^x \)[/tex]
Sum of squared errors (SSE): 14938.16
By comparing the SSE values, we see that Model 2, [tex]\( f(x) = 5(0.2)^x \)[/tex], has the smallest sum of squared errors (13.010000000000085). Therefore, this model best represents the given data.
The best exponential model for the data is:
[tex]\[ f(x) = 5(0.2)^x \][/tex]
