To determine which type of function best models the data given in the table, we need to carefully analyze the given [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values:
\begin{center}
\begin{tabular}{|l|l|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
0 & 3 \\
\hline
1 & 6 \\
\hline
2 & 12 \\
\hline
3 & 24 \\
\hline
4 & 48 \\
\hline
\end{tabular}
\end{center}
### Step-by-Step Analysis
1. Examining the pattern:
- Check for a linear pattern:
- Calculate first differences [tex]\( \Delta y \)[/tex] (y[i+1] - y[i]):
- [tex]\(6 - 3 = 3\)[/tex]
- [tex]\(12 - 6 = 6\)[/tex]
- [tex]\(24 - 12 = 12\)[/tex]
- [tex]\(48 - 24 = 24\)[/tex]
The differences are not constant, so the function is not linear.
- Check for a quadratic pattern:
- A quadratic function would result in second differences (differences of differences) being constant.
Second differences:
- [tex]\(6 - 3 = 3\)[/tex]
- [tex]\(12 - 6 = 6\)[/tex]
- [tex]\(24 - 12 = 12\)[/tex]
- [tex]\(48 - 24 = 24\)[/tex]
- Differences of differences:
- [tex]\(6 - 3 = 3\)[/tex]
- [tex]\(12 - 6 = 6\)[/tex]
- [tex]\(24 - 12 = 12\)[/tex]
The second differences are not constant, so the function is not quadratic.
- Check for an exponential pattern:
- Calculate the ratios [tex]\( \frac{y[i+1]}{y[i]} \)[/tex] (y[i+1] / y[i]):
- [tex]\( \frac{6}{3} = 2 \)[/tex]
- [tex]\( \frac{12}{6} = 2 \)[/tex]
- [tex]\( \frac{24}{12} = 2 \)[/tex]
- [tex]\( \frac{48}{24} = 2 \)[/tex]
The ratios are constant, which suggests an exponential function.
2. Finding the function parameters:
An exponential function has the form [tex]\( y = a(b)^x \)[/tex]:
- From the data, [tex]\( y = 3 \)[/tex] when [tex]\( x = 0 \)[/tex], so [tex]\( a = 3 \)[/tex].
- The common ratio [tex]\( b \)[/tex] is determined from the first ratio:
- [tex]\( b = \frac{6}{3} = 2 \)[/tex]
3. Conclusion:
The function that models the data is an exponential function given by:
[tex]\[
y = 3 \cdot 2^x
\][/tex]
Thus, the function that models the data in the table is:
[tex]\[
y = 3 \cdot 2^x
\][/tex]
