Answer :
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]
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]
