To determine the corresponding values of [tex]\( g(x) = -3 \left(\frac{1}{2}\right)^x \)[/tex] for the given [tex]\( x \)[/tex]-values, we'll evaluate the function at each [tex]\( x \)[/tex]-value and then plot the points on the graph.
1. Calculate [tex]\( g(x) \)[/tex] for each [tex]\( x \)[/tex]-value:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) = -3 \left(\frac{1}{2}\right)^{-2} = -3 \left(2\right)^2 = -3 \cdot 4 = -12
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = -3 \left(\frac{1}{2}\right)^{-1} = -3 \left(2\right) = -3 \cdot 2 = -6
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = -3 \left(\frac{1}{2}\right)^0 = -3 \cdot 1 = -3
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
g(1) = -3 \left(\frac{1}{2}\right)^1 = -3 \cdot \frac{1}{2} = -1.5
\][/tex]
2. Fill in the table with the calculated values:
\begin{tabular}{|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -2 & -1 & 0 & 1 \\
\hline
[tex]$g(x)$[/tex] & -12 & -6 & -3 & -1.5 \\
\hline
\end{tabular}
3. Plot each point [tex]\((x, g(x))\)[/tex] on the graph:
- Plot the point [tex]\((-2, -12)\)[/tex]
- Plot the point [tex]\((-1, -6)\)[/tex]
- Plot the point [tex]\((0, -3)\)[/tex]
- Plot the point [tex]\((1, -1.5)\)[/tex]
Now you've successfully determined the values of the function [tex]\( g(x) = -3 \left(\frac{1}{2}\right)^x \)[/tex] at the given [tex]\( x \)[/tex]-values and plotted them on the graph!
