To determine the multiplicative rate of change of the exponential function represented by the given table, let's first understand what this rate is and then confirm it through the steps.
The multiplicative rate of change for an exponential function is often referred to as the "base" of the exponential function when written in the form [tex]\( y = ab^x \)[/tex]. This rate tells us how [tex]\( y \)[/tex] changes as [tex]\( x \)[/tex] increases.
Here’s a detailed step-by-step approach to finding the multiplicative rate of change:
1. Identify the values:
From the table:
- When [tex]\( x = 1 \)[/tex], [tex]\( y = 6 \)[/tex]
- When [tex]\( x = 2 \)[/tex], [tex]\( y = 4 \)[/tex]
- When [tex]\( x = 3 \)[/tex], [tex]\( y = \frac{8}{3} \)[/tex]
- When [tex]\( x = 4 \)[/tex], [tex]\( y = \frac{16}{9} \)[/tex]
2. Calculate the rate of change between consecutive [tex]\( y \)[/tex]-values:
a. From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{y_2}{y_1} = \frac{4}{6} = \frac{2}{3}
\][/tex]
b. From [tex]\( x = 2 \)[/tex] to [tex]\( x = 3 \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{y_3}{y_2} = \frac{\frac{8}{3}}{4} = \frac{\frac{8}{3}}{\frac{12}{3}} = \frac{8}{12} = \frac{2}{3}
\][/tex]
c. From [tex]\( x = 3 \)[/tex] to [tex]\( x = 4 \)[/tex]:
[tex]\[
\text{Rate of change} = \frac{y_4}{y_3} = \frac{\frac{16}{9}}{\frac{8}{3}} = \frac{\frac{16}{9}}{\frac{24}{9}} = \frac{16}{24} = \frac{2}{3}
\][/tex]
3. Check for consistency in the rate of change:
The rate of change for all steps is [tex]\(\frac{2}{3}\)[/tex].
4. Conclusion:
The consistent multiplicative rate of change is [tex]\(\frac{2}{3}\)[/tex].
Thus, the multiplicative rate of change of the function is [tex]\(\frac{2}{3}\)[/tex]. The correct answer is:
[tex]\[
\boxed{\frac{2}{3}}
\][/tex]
