To determine the multiplicative rate of change for the given exponential function represented by the table, let’s analyze the values step-by-step.
Firstly, the table values are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 6 \\
\hline
2 & 4 \\
\hline
3 & \frac{8}{3} \\
\hline
4 & \frac{16}{9} \\
\hline
\end{array}
\][/tex]
Exponential functions have the form [tex]\( y = ab^x \)[/tex], where [tex]\( b \)[/tex] represents the multiplicative rate of change.
Now, check the rate of change between each consecutive value of [tex]\( y \)[/tex]:
1. From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{y(2)}{y(1)} = \frac{4}{6} = \frac{2}{3}
\][/tex]
2. From [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{y(3)}{y(2)} = \frac{\frac{8}{3}}{4} = \frac{8}{3} \cdot \frac{1}{4} = \frac{8}{12} = \frac{2}{3}
\][/tex]
3. From [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{y(4)}{y(3)} = \frac{\frac{16}{9}}{\frac{8}{3}} = \frac{16}{9} \cdot \frac{3}{8} = \frac{48}{72} = \frac{2}{3}
\][/tex]
The ratio is consistent and equals [tex]\(\frac{2}{3}\)[/tex] for each pair of consecutive [tex]\( y \)[/tex] values.
Thus, the multiplicative rate of change of the function is:
[tex]\[
\boxed{\frac{2}{3}}
\][/tex]
