which table represents an exponential function of the form y = b^x when 0 < b < 1?

Answer:

The second - table (with values: $x=-3,y = 27;x=-2,y = 9;x=-1,y = 3;x = 0,y = 1;x = 1,y=\frac{1}{3};x = 2,y=\frac{1}{9};x = 3,y=\frac{1}{27}$)

Explanation:

Step1: Recall exponential function properties

For $y = b^{x}$ with $0 < b<1$, as $x$ increases, $y$ decreases. Also, when $x = 0$, $y=1$.

Step2: Analyze the first table

In the first table, as $x$ increases from $- 3$ to $3$, $y$ first decreases from $x=-3$ to $x = 0$ and then increases from $x = 0$ to $x = 3$, so it's not of the form $y = b^{x}$ with $0 < b<1$.

Step3: Analyze the second table

In the second table, when $x = 0$, $y = 1$. As $x$ increases from $0$ to $3$, $y$ decreases ($y$ goes from $1$ to $\frac{1}{27}$), and as $x$ decreases from $0$ to $-3$, $y$ increases ($y$ goes from $1$ to $27$), which is consistent with $y = b^{x}$ where $0 < b<1$.

Step4: Analyze the third and fourth tables

The third and fourth tables have non - integer $x$ values and also have negative $y$ values in some cases. For the function $y = b^{x}$ ($b>0$), $y>0$ for all real $x$. So they do not represent $y = b^{x}$ with $0 < b<1$.