To determine whether the function represented by the given table is an exponential growth or decay, we need to examine the ratios of consecutive [tex]\( y \)[/tex]-values. Consistent ratios imply exponential behavior.
Given the table:
[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]
Calculate the ratios of consecutive [tex]\( y \)[/tex]-values:
1. [tex]\(\frac{48}{192} = 0.25\)[/tex]
2. [tex]\(\frac{12}{48} = 0.25\)[/tex]
3. [tex]\(\frac{3}{12} = 0.25\)[/tex]
4. [tex]\(\frac{3/4}{3} = 0.25\)[/tex]
5. [tex]\(\frac{3/16}{3/4} = 0.25\)[/tex]
6. [tex]\(\frac{3/64}{3/16} = 0.25\)[/tex]
The ratios are all equal to 0.25.
Since all the ratios are 0.25, and each ratio is less than 1, the function represents an exponential decay.
Thus, the function in the table represents decay.
