To determine the multiplicative rate of change of the exponential function represented by the table, we need to identify how the [tex]\( y \)[/tex]-values change as the [tex]\( x \)[/tex]-values increase.
The values given in the table are:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 0.25 \\
\hline
2 & 0.125 \\
\hline
3 & 0.0625 \\
\hline
4 & 0.03125 \\
\hline
\end{array}
\][/tex]
The multiplicative rate of change in an exponential function can be found by calculating the ratio of consecutive [tex]\( y \)[/tex]-values.
1. For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{y_2}{y_1} = \frac{0.125}{0.25} = 0.5
\][/tex]
2. For [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{y_3}{y_2} = \frac{0.0625}{0.125} = 0.5
\][/tex]
3. For [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{y_4}{y_3} = \frac{0.03125}{0.0625} = 0.5
\][/tex]
We observe that the rate of change is consistent across all intervals. Since it is constant, we confirm that the multiplicative rate of change for the function is:
[tex]\[
\boxed{0.5}
\][/tex]
