the table represents an exponential function. what is the multiplicative rate of change of the…

Explanation:

Step1: Identify consecutive y-values from the table.

The table provides points $(x, y)$. For an exponential function, the multiplicative rate of change is the ratio of consecutive $y$-values when $x$ increases by 1. Let $y_1$ be the $y$-value when $x=1$, and $y_2$ be the $y$-value when $x=2$. From the table: $y_1 = \frac{3}{2}$ (for $x=1$) $y_2 = \frac{9}{8}$ (for $x=2$)

Step2: Calculate the ratio of the second y-value to the first y-value.

The multiplicative rate of change $b$ is given by $b = \frac{y_2}{y_1}$. $$b = \frac{\frac{9}{8}}{\frac{3}{2}}$$

Step3: Simplify the expression for the rate of change.

To divide fractions, multiply by the reciprocal of the divisor. $$b = \frac{9}{8} \times \frac{2}{3}$$ $$b = \frac{9 \times 2}{8 \times 3}$$ $$b = \frac{18}{24}$$

Step4: Reduce the fraction to its simplest form.

Divide both the numerator and the denominator by their greatest common divisor, which is 6. $$b = \frac{18 \div 6}{24 \div 6}$$ $$b = \frac{3}{4}$$ To verify, we can use another pair of consecutive y-values, for example, $y_3 = \frac{27}{32}$ (for $x=3$) and $y_2 = \frac{9}{8}$ (for $x=2$). $$b = \frac{y_3}{y_2} = \frac{\frac{27}{32}}{\frac{9}{8}} = \frac{27}{32} \times \frac{8}{9} = \frac{27 \times 8}{32 \times 9} = \frac{216}{288}$$ Simplifying $\frac{216}{288}$ by dividing by their greatest common divisor, 72: $$b = \frac{216 \div 72}{288 \div 72} = \frac{3}{4}$$ The multiplicative rate of change is consistent.

Answer:

B. $\frac{3}{4}$