To determine which table models exponential growth, we need to examine how the [tex]\( y \)[/tex]-values change with respect to the [tex]\( x \)[/tex]-values. Exponential growth typically follows the form [tex]\( y = a \cdot b^x \)[/tex], where [tex]\( a \)[/tex] is the initial value (when [tex]\( x = 0 \)[/tex]), and [tex]\( b \)[/tex] is the growth factor.
Let's analyze each table:
1. First table:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline 0 & 1 \\
\hline 1 & 4 \\
\hline 2 & 16 \\
\hline
\end{array}
\][/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 4 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 16 \)[/tex]
Let's check the ratios:
[tex]\[
\frac{y_1}{y_0} = \frac{4}{1} = 4, \quad \frac{y_2}{y_1} = \frac{16}{4} = 4
\][/tex]
Since the ratio is constant (4), this suggests exponential growth with a growth factor [tex]\( b = 4 \)[/tex].
2. Second table:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline -1 & 4 \\
\hline 0 & 1 \\
\hline 1 & \frac{1}{4} \\
\hline
\end{array}
\][/tex]
- When [tex]\( x = -1 \)[/tex], [tex]\( y = 4 \)[/tex]
- When [tex]\( x = 0 \)[/tex], [tex]\( y = 1 \)[/tex]
- When [tex]\( x = 1 \)[/tex], [tex]\( y = \frac{1}{4} \)[/tex]
Let's check the ratios:
[tex]\[
\frac{y_1}{y_0} = \frac{1}{4} = \frac{1}{4}, \quad \frac{y_2}{y_1} = \frac{\frac{1}{4}}{1} = \frac{1}{4}
\][/tex]
Since the ratio is constant ([tex]\(\frac{1}{4}\)[/tex]), this suggests exponential decay with a decay factor [tex]\( b = \frac{1}{4} \)[/tex], rather than growth.
3. Third table:
[tex]\[
\begin{array}{|c|c|}
\hline x & y \\
\hline -2 & -4 \\
\hline 0 & 0 \\
\hline 2 & 4 \\
\hline
\end{array}
\][/tex]
This set of values does not fit an exponential growth model because the [tex]\( y \)[/tex]-values include negative numbers and zero. Exponential functions do not take negative values or zero for non-complex numbers.
4. "None of these":
Examination of all three tables shows that only the first table adheres to a consistent exponential growth pattern.
Therefore, the table that models exponential growth is:
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
0 & 1 \\
\hline
1 & 4 \\
\hline
2 & 16 \\
\hline
\end{tabular}
