To determine the type of function [tex]\( B(x) \)[/tex], whether it is linear, quadratic, or exponential, we will analyze the given data points across several criteria:
1. Ratios to check for exponential growth.
2. First differences to check for linearity.
3. Second differences to check for quadratic characteristics.
The given data points are:
[tex]\[ \begin{aligned}
&\text{Years} \ (x): & 0 & 1 & 2 & 3 & 4 & 5 \\
& B(x): & 2 & 6 & 18 & 54 & 162 & 486 \\
\end{aligned} \][/tex]
### Step 1: Calculate Ratios
We calculate the ratios [tex]\(\frac{B(x+1)}{B(x)}\)[/tex]:
[tex]\[
\begin{aligned}
&\frac{6}{2} = 3.0, \\
&\frac{18}{6} = 3.0, \\
&\frac{54}{18} = 3.0, \\
&\frac{162}{54} = 3.0, \\
&\frac{486}{162} = 3.0.
\end{aligned}
\][/tex]
The ratios are all consistent and equal to 3.0. This consistency in ratios is indicative of an exponential function.
### Step 2: Calculate First Differences
We calculate the first differences [tex]\( B(x+1) - B(x) \)[/tex]:
[tex]\[
\begin{aligned}
&6 - 2 = 4, \\
&18 - 6 = 12, \\
&54 - 18 = 36, \\
&162 - 54 = 108, \\
&486 - 162 = 324.
\end{aligned}
\][/tex]
The first differences are:
[tex]\[ [4, 12, 36, 108, 324] \][/tex]
These differences are not constant, which means that the function is not linear.
### Step 3: Calculate Second Differences
We calculate the second differences by taking the differences of the first differences:
[tex]\[
\begin{aligned}
&12 - 4 = 8, \\
&36 - 12 = 24, \\
&108 - 36 = 72, \\
&324 - 108 = 216.
\end{aligned}
\][/tex]
The second differences are:
[tex]\[ [8, 24, 72, 216] \][/tex]
These second differences are also not constant, indicating the function is not quadratic.
### Conclusion
Based on our calculations:
- The ratios [tex]\(\frac{B(x+1)}{B(x)}\)[/tex] are constant (3.0), suggesting an exponential function.
- The first differences are not constant, ruling out a linear function.
- The second differences are not constant, ruling out a quadratic function.
Thus, we conclude that [tex]\( B(x) \)[/tex] is an exponential function.
[tex]\[
\begin{array}{|r|r|r|r|r|}
\hline
\text{Years (x)} & \text{Batana } B(x) & \text{Ratios} & \text{First Differences} & \text{Second Differences} \\
\hline
0 & 2 & & & \\
\hline
1 & 6 & 3.0 & & \\
\hline
2 & 18 & 3.0 & 4 & \\
\hline
3 & 54 & 3.0 & 12 & 8 \\
\hline
4 & 162 & 3.0 & 36 & 24 \\
\hline
5 & 486 & 3.0 & 108 & 72 \\
\hline
& & & 324 & 216 \\
\hline
\end{array}
\][/tex]
