Answer :
Let's analyze each table to see if the values represent an exponential function. An exponential function grows by constant ratios rather than by constant differences.
Table A:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 5 \\ \hline 2 & 10 \\ \hline 3 & 20 \\ \hline 4 & 40 \\ \hline 5 & 80 \\ \hline \end{tabular} \][/tex]
To check if these values represent an exponential function, we look for a constant ratio between consecutive [tex]\( y \)[/tex] values:
- [tex]\( \frac{10}{5} = 2 \)[/tex]
- [tex]\( \frac{20}{10} = 2 \)[/tex]
- [tex]\( \frac{40}{20} = 2 \)[/tex]
- [tex]\( \frac{80}{40} = 2 \)[/tex]
Each ratio is 2, indicating the values in Table A follow an exponential pattern.
Table B:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 5 \\ \hline 2 & 9 \\ \hline 3 & 13 \\ \hline 4 & 17 \\ \hline 5 & 21 \\ \hline \end{tabular} \][/tex]
To check if these values represent an exponential function, we look for a constant ratio between consecutive [tex]\( y \)[/tex] values:
- [tex]\( \frac{9}{5} = 1.8 \)[/tex]
- [tex]\( \frac{13}{9} \approx 1.44 \)[/tex]
- [tex]\( \frac{17}{13} \approx 1.31 \)[/tex]
- [tex]\( \frac{21}{17} \approx 1.24 \)[/tex]
The ratios are not constant. Moreover, if we look for a constant difference:
- [tex]\( 9 - 5 = 4\)[/tex]
- [tex]\( 13 - 9 = 4\)[/tex]
- [tex]\( 17 - 13 = 4\)[/tex]
- [tex]\( 21 - 17 = 4\)[/tex]
There is a constant difference of 4, indicating these values follow a linear pattern, not exponential.
Table C:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 16 \\ \hline 2 & 29 \\ \hline 3 & 42 \\ \hline 4 & 55 \\ \hline \end{tabular} \][/tex]
To check if these values represent an exponential function, we look for a constant ratio between consecutive [tex]\( y \)[/tex] values:
- [tex]\( \frac{29}{16} \approx 1.81 \)[/tex]
- [tex]\( \frac{42}{29} \approx 1.45 \)[/tex]
- [tex]\( \frac{55}{42} \approx 1.31 \)[/tex]
The ratios are not constant. Moreover, if we look for a constant difference:
- [tex]\( 29 - 16 = 13 \)[/tex]
- [tex]\( 42 - 29 = 13 \)[/tex]
- [tex]\( 55 - 42 = 13 \)[/tex]
There is a constant difference of 13, indicating these values follow a linear pattern, not exponential.
Based on the analysis, the values in Table A represent an exponential function because they have a constant ratio between consecutive values.
Table A:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 5 \\ \hline 2 & 10 \\ \hline 3 & 20 \\ \hline 4 & 40 \\ \hline 5 & 80 \\ \hline \end{tabular} \][/tex]
To check if these values represent an exponential function, we look for a constant ratio between consecutive [tex]\( y \)[/tex] values:
- [tex]\( \frac{10}{5} = 2 \)[/tex]
- [tex]\( \frac{20}{10} = 2 \)[/tex]
- [tex]\( \frac{40}{20} = 2 \)[/tex]
- [tex]\( \frac{80}{40} = 2 \)[/tex]
Each ratio is 2, indicating the values in Table A follow an exponential pattern.
Table B:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 5 \\ \hline 2 & 9 \\ \hline 3 & 13 \\ \hline 4 & 17 \\ \hline 5 & 21 \\ \hline \end{tabular} \][/tex]
To check if these values represent an exponential function, we look for a constant ratio between consecutive [tex]\( y \)[/tex] values:
- [tex]\( \frac{9}{5} = 1.8 \)[/tex]
- [tex]\( \frac{13}{9} \approx 1.44 \)[/tex]
- [tex]\( \frac{17}{13} \approx 1.31 \)[/tex]
- [tex]\( \frac{21}{17} \approx 1.24 \)[/tex]
The ratios are not constant. Moreover, if we look for a constant difference:
- [tex]\( 9 - 5 = 4\)[/tex]
- [tex]\( 13 - 9 = 4\)[/tex]
- [tex]\( 17 - 13 = 4\)[/tex]
- [tex]\( 21 - 17 = 4\)[/tex]
There is a constant difference of 4, indicating these values follow a linear pattern, not exponential.
Table C:
[tex]\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 1 & 16 \\ \hline 2 & 29 \\ \hline 3 & 42 \\ \hline 4 & 55 \\ \hline \end{tabular} \][/tex]
To check if these values represent an exponential function, we look for a constant ratio between consecutive [tex]\( y \)[/tex] values:
- [tex]\( \frac{29}{16} \approx 1.81 \)[/tex]
- [tex]\( \frac{42}{29} \approx 1.45 \)[/tex]
- [tex]\( \frac{55}{42} \approx 1.31 \)[/tex]
The ratios are not constant. Moreover, if we look for a constant difference:
- [tex]\( 29 - 16 = 13 \)[/tex]
- [tex]\( 42 - 29 = 13 \)[/tex]
- [tex]\( 55 - 42 = 13 \)[/tex]
There is a constant difference of 13, indicating these values follow a linear pattern, not exponential.
Based on the analysis, the values in Table A represent an exponential function because they have a constant ratio between consecutive values.
