Answer :
To determine which table of values models exponential growth, we must check the ratio between consecutive terms in each sequence. Exponential growth is characterized by a constant ratio between consecutive terms. Let's analyze each option step-by-step:
### Option (a)
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline[tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline[tex]$f(x)$[/tex] & 15 & 30 & 45 & 60 \\
\hline
\end{tabular}
\][/tex]
- Ratio of \( \frac{30}{15} = 2.0 \)
- Ratio of \( \frac{45}{30} = 1.5 \)
- Ratio of \( \frac{60}{45} = 1.33...( \approx 1.333) \)
The ratios are not constant, so this is not exponential growth.
### Option (b)
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline[tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline[tex]$f(x)$[/tex] & 15 & 30 & 60 & 120 \\
\hline
\end{tabular}
\][/tex]
- Ratio of \( \frac{30}{15} = 2.0 \)
- Ratio of \( \frac{60}{30} = 2.0 \)
- Ratio of \( \frac{120}{60} = 2.0 \)
The ratios are constant, so this sequence demonstrates exponential growth.
### Option (c)
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline[tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline[tex]$f(x)$[/tex] & 120 & 60 & 30 & 15 \\
\hline
\end{tabular}
\][/tex]
- Ratio of \( \frac{60}{120} = 0.5 \)
- Ratio of \( \frac{30}{60} = 0.5 \)
- Ratio of \( \frac{15}{30} = 0.5 \)
Even though the ratios are constant, they represent decay rather than growth. This models exponential decay, not exponential growth.
### Option (d)
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline [tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline [tex]$f (x)$[/tex] & 60 & 45 & 30 & 15 \\
\hline
\end{tabular}
\][/tex]
- Ratio of \( \frac{45}{60} = 0.75 \)
- Ratio of \( \frac{30}{45} = 0.666... ( \approx 0.667) \)
- Ratio of \( \frac{15}{30} = 0.5 \)
The ratios are not constant, so this is not exponential growth.
Therefore, the table of values that models exponential growth is:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline[tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline[tex]$f(x)$[/tex] & 15 & 30 & 60 & 120 \\
\hline
\end{tabular}
\][/tex]
Thus, the correct option is (b).
### Option (a)
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline[tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline[tex]$f(x)$[/tex] & 15 & 30 & 45 & 60 \\
\hline
\end{tabular}
\][/tex]
- Ratio of \( \frac{30}{15} = 2.0 \)
- Ratio of \( \frac{45}{30} = 1.5 \)
- Ratio of \( \frac{60}{45} = 1.33...( \approx 1.333) \)
The ratios are not constant, so this is not exponential growth.
### Option (b)
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline[tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline[tex]$f(x)$[/tex] & 15 & 30 & 60 & 120 \\
\hline
\end{tabular}
\][/tex]
- Ratio of \( \frac{30}{15} = 2.0 \)
- Ratio of \( \frac{60}{30} = 2.0 \)
- Ratio of \( \frac{120}{60} = 2.0 \)
The ratios are constant, so this sequence demonstrates exponential growth.
### Option (c)
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline[tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline[tex]$f(x)$[/tex] & 120 & 60 & 30 & 15 \\
\hline
\end{tabular}
\][/tex]
- Ratio of \( \frac{60}{120} = 0.5 \)
- Ratio of \( \frac{30}{60} = 0.5 \)
- Ratio of \( \frac{15}{30} = 0.5 \)
Even though the ratios are constant, they represent decay rather than growth. This models exponential decay, not exponential growth.
### Option (d)
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline [tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline [tex]$f (x)$[/tex] & 60 & 45 & 30 & 15 \\
\hline
\end{tabular}
\][/tex]
- Ratio of \( \frac{45}{60} = 0.75 \)
- Ratio of \( \frac{30}{45} = 0.666... ( \approx 0.667) \)
- Ratio of \( \frac{15}{30} = 0.5 \)
The ratios are not constant, so this is not exponential growth.
Therefore, the table of values that models exponential growth is:
[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline[tex]$x$[/tex] & 0 & 1 & 2 & 3 \\
\hline[tex]$f(x)$[/tex] & 15 & 30 & 60 & 120 \\
\hline
\end{tabular}
\][/tex]
Thus, the correct option is (b).
