To determine the multiplicative rate of change for the given exponential function, let's analyze the relationship between the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] values given in the table.
First, we observe the values:
[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]
We need to find the rate of change by looking at the successive terms of [tex]\( y \)[/tex]. This rate of change can be calculated by dividing a [tex]\( y \)[/tex]-value by the previous [tex]\( y \)[/tex]-value.
Let's compute it step-by-step:
1. Calculate the ratio of [tex]\( y \)[/tex] for [tex]\( x = 2 \)[/tex] to [tex]\( y \)[/tex] for [tex]\( x = 1 \)[/tex]:
[tex]\[
\frac{y(2)}{y(1)} = \frac{0.125}{0.25} = 0.5
\][/tex]
2. Verify consistency by calculating the ratio for the next pairs of [tex]\( y \)[/tex]-values:
[tex]\[
\frac{y(3)}{y(2)} = \frac{0.0625}{0.125} = 0.5
\][/tex]
[tex]\[
\frac{y(4)}{y(3)} = \frac{0.03125}{0.0625} = 0.5
\][/tex]
As we can see, the ratio is consistently [tex]\( 0.5 \)[/tex] for each pair of successive [tex]\( y \)[/tex]-values.
Thus, the multiplicative rate of change of the function is:
[tex]\[
0.5
\][/tex]
So, the correct answer is:
- 0.5
