Let's analyze each table to determine whether the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] is linear or exponential.
### Function A:
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
1 & 2 \\
\hline
2 & 4 \\
\hline
3 & 8 \\
\hline
4 & 16 \\
\hline
\end{tabular}
To identify if this is linear or exponential:
- An exponential function involves a constant multiplicative rate between points.
- Checking ratios:
[tex]\[
\frac{4}{2} = 2, \quad \frac{8}{4} = 2, \quad \frac{16}{8} = 2
\][/tex]
Since the ratios between successive terms are constant (all equal to 2), this indicates an exponential relationship.
Function A is exponential.
### Function B:
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
2 & 1.44 \\
\hline
4 & 3.6 \\
\hline
6 & 9 \\
\hline
8 & 22.5 \\
\hline
\end{tabular}
To identify if this is linear or exponential:
- An exponential function involves a constant multiplicative rate between points.
- Checking ratios:
[tex]\[
\frac{3.6}{1.44} = 2.5, \quad \frac{9}{3.6} = 2.5, \quad \frac{22.5}{9} = 2.5
\][/tex]
Since the ratios between successive terms are constant (all equal to 2.5), this indicates an exponential relationship.
Function B is exponential.
### Function C:
\begin{tabular}{|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline
2 & 5 \\
\hline
5 & 12.5 \\
\hline
8 & 20 \\
\hline
11 & 27.5 \\
\hline
\end{tabular}
To identify if this is linear or exponential:
- A linear function involves a constant additive rate between points.
- Checking differences:
[tex]\[
12.5 - 5 = 7.5, \quad 20 - 12.5 = 7.5, \quad 27.5 - 20 = 7.5
\][/tex]
Since the differences between successive terms are constant (all equal to 7.5), this indicates a linear relationship.
Function C is linear.
So, in conclusion:
- Function A is exponential.
- Function B is exponential.
- Function C is linear.
