To determine whether the function in the table represents exponential growth or decay, we need to analyze the ratios between successive [tex]\( y \)[/tex] values. These ratios will help us identify the base of the exponential function.
Let's calculate these ratios step-by-step.
1. We have the data points:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|}
\hline
x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\
\hline
y & 192 & 48 & 12 & 3 & \frac{3}{4} & \frac{3}{16} & \frac{3}{64} \\
\hline
\end{array}
\][/tex]
2. Calculate the ratios between successive [tex]\( y \)[/tex] values:
[tex]\[
\text{Ratio between } y(-2) \text{ and } y(-3): \frac{y(-2)}{y(-3)} = \frac{48}{192} = 0.25
\][/tex]
[tex]\[
\text{Ratio between } y(-1) \text{ and } y(-2): \frac{y(-1)}{y(-2)} = \frac{12}{48} = 0.25
\][/tex]
[tex]\[
\text{Ratio between } y(0) \text{ and } y(-1): \frac{y(0)}{y(-1)} = \frac{3}{12} = 0.25
\][/tex]
[tex]\[
\text{Ratio between } y(1) \text{ and } y(0): \frac{y(1)}{y(0)} = \frac{0.75}{3} = 0.25
\][/tex]
[tex]\[
\text{Ratio between } y(2) \text{ and } y(1): \frac{y(2)}{y(1)} = \frac{\frac{3}{16}}{\frac{3}{4}} = 0.25
\][/tex]
[tex]\[
\text{Ratio between } y(3) \text{ and } y(2): \frac{y(3)}{y(2)} = \frac{\frac{3}{64}}{\frac{3}{16}} = 0.25
\][/tex]
3. We see that the ratio is consistent across all pairs and equals 0.25. This means that the function can be defined with a base of 0.25 in the exponential function form.
Since the base of the exponential function is [tex]\( 0.25 \)[/tex], which is less than 1, this indicates that the function represents exponential decay.
Therefore, the correct statement is:
[tex]\[ \boxed{\text{The function represents exponential decay because the base equals } \frac{1}{4}.} \][/tex]
