To determine the multiplicative rate of change of the given exponential function, we need to examine how the value of [tex]\(y\)[/tex] changes as [tex]\(x\)[/tex] increases. Let’s go through the steps systematically:
1. Identify the given data points:
[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]
2. Calculate the multiplicative rate of change between consecutive [tex]\(y\)[/tex]-values:
The rate of change between two consecutive points [tex]\((x_i, y_i)\)[/tex] and [tex]\((x_{i+1}, y_{i+1})\)[/tex] in an exponential function should be consistent. Therefore, we compute:
[tex]\[
\text{Rate of change from } x = 1 \text{ to } x = 2: \frac{y_2}{y_1} = \frac{4}{6} = \frac{2}{3}
\][/tex]
[tex]\[
\text{Rate of change from } x = 2 \text{ to } x = 3: \frac{y_3}{y_2} = \frac{\frac{8}{3}}{4} = \frac{8}{3} \cdot \frac{1}{4} = \frac{8}{12} = \frac{2}{3}
\][/tex]
[tex]\[
\text{Rate of change from } x = 3 \text{ to } x = 4: \frac{y_4}{y_3} = \frac{\frac{16}{9}}{\frac{8}{3}} = \frac{16}{9} \cdot \frac{3}{8} = \frac{48}{72} = \frac{2}{3}
\][/tex]
3. Evaluate the consistency of the rate of change:
All calculated rates of change are [tex]\(\frac{2}{3}\)[/tex], indicating a consistent multiplicative rate of change, as expected for an exponential function.
4. Conclusion:
The multiplicative rate of change for the given exponential function is:
[tex]\[
\boxed{\frac{2}{3}}
\][/tex]
