What is the multiplicative rate of change of the function represented in the table?

\begin{tabular}{|c|c|}
\hline [tex]$x$[/tex] & [tex]$y$[/tex] \\
\hline 1 & 0.25 \\
\hline 2 & 0.125 \\
\hline 3 & 0.0625 \\
\hline 4 & 0.03125 \\
\hline
\end{tabular}

A. 0.2
B. 0.25
C. 0.5
D. 0.75



Answer :

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]