Expand your horizons with the diverse and informative answers found on IDNLearn.com. Get the information you need from our community of experts, who provide detailed and trustworthy answers.
Sagot :
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]
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. Your questions find clarity at IDNLearn.com. Thanks for stopping by, and come back for more dependable solutions.