To graph the function [tex]\( f(x) = \left( \frac{1}{2} \right)^x \)[/tex], we first need to evaluate the function for different values of [tex]\( x \)[/tex]. We pick the values [tex]\( x = -2, -1, 0, 1, \)[/tex] and [tex]\( 2 \)[/tex].
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = \left( \frac{1}{2} \right)^{-2} = (2)^2 = 4 \][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = \left( \frac{1}{2} \right)^{-1} = 2 \][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = \left( \frac{1}{2} \right)^0 = 1 \][/tex]
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = \left( \frac{1}{2} \right)^1 = \frac{1}{2} \][/tex]
5. For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \][/tex]
Now, we can complete the table of coordinates:
\begin{tabular}{|c|c|c|c|c|c|}
\hline [tex]$x$[/tex] & -2 & -1 & 0 & 1 & 2 \\
\hline [tex]$y$[/tex] & 4 & 2 & 1 & 0.5 & 0.25 \\
\hline
\end{tabular}
These coordinates can be used to graph the function [tex]\( f(x) = \left( \frac{1}{2} \right)^x \)[/tex].
