To determine the multiplicative rate of change of the given exponential function, we need to examine how the [tex]\( y \)[/tex]-values change as the [tex]\( x \)[/tex]-values increase. The given [tex]\( x \)[/tex]-values are 1, 2, 3, and 4, with corresponding [tex]\( y \)[/tex]-values of 0.25, 0.125, 0.0625, and 0.03125.
An exponential function has a consistent multiplicative rate of change, meaning that the ratio between successive [tex]\( y \)[/tex]-values remains constant.
Let's find the ratios between successive [tex]\( y \)[/tex]-values:
1. The ratio between the [tex]\( y \)[/tex]-value for [tex]\( x = 2 \)[/tex] and the [tex]\( y \)[/tex]-value for [tex]\( x = 1 \)[/tex]:
[tex]\[
\text{Rate of change}_1 = \frac{y (2)}{y (1)} = \frac{0.125}{0.25} = 0.5
\][/tex]
2. The ratio between the [tex]\( y \)[/tex]-value for [tex]\( x = 3 \)[/tex] and the [tex]\( y \)[/tex]-value for [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Rate of change}_2 = \frac{y (3)}{y (2)} = \frac{0.0625}{0.125} = 0.5
\][/tex]
3. The ratio between the [tex]\( y \)[/tex]-value for [tex]\( x = 4 \)[/tex] and the [tex]\( y \)[/tex]-value for [tex]\( x = 3 \)[/tex]:
[tex]\[
\text{Rate of change}_3 = \frac{y (4)}{y (3)} = \frac{0.03125}{0.0625} = 0.5
\][/tex]
Since the ratio remains consistent at 0.5 for each comparison, the multiplicative rate of change of the function is:
[tex]\[
\boxed{0.5}
\][/tex]
