Answer :
To determine the corresponding values for [tex]\( g(x) = -3\left(\frac{1}{2}\right)^x \)[/tex] for the given [tex]\( x \)[/tex]-values, we calculate as follows:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = -3\left(\frac{1}{2}\right)^{-2} = -3 \times 2^2 = -3 \times 4 = -12 \][/tex]
Therefore, [tex]\( g(-2) = -12 \)[/tex].
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = -3\left(\frac{1}{2}\right)^{-1} = -3 \times 2 = -6 \][/tex]
Therefore, [tex]\( g(-1) = -6 \)[/tex].
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -3 \left(\frac{1}{2}\right)^0 = -3 \times 1 = -3 \][/tex]
Therefore, [tex]\( g(0) = -3 \)[/tex].
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -3\left(\frac{1}{2}\right)^1 = -3 \times \frac{1}{2} = -1.5 \][/tex]
Therefore, [tex]\( g(1) = -1.5 \)[/tex].
So, the table with the corresponding values filled in is:
\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}
### Graphing the Points
The points to be plotted on the graph are:
- [tex]\((-2, -12)\)[/tex]
- [tex]\((-1, -6)\)[/tex]
- [tex]\((0, -3)\)[/tex]
- [tex]\((1, -1.5)\)[/tex]
On a coordinate plane:
- The x-axis should have values at least from [tex]\(-2\)[/tex] to [tex]\(1\)[/tex].
- The y-axis should accommodate values at least from [tex]\(-12\)[/tex] to [tex]\(0\)[/tex].
Once you plot these points, you will be able to visualize the function [tex]\( g(x) = -3\left(\frac{1}{2}\right)^x \)[/tex]. Ensure the points are accurately placed according to the values calculated.
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[ g(-2) = -3\left(\frac{1}{2}\right)^{-2} = -3 \times 2^2 = -3 \times 4 = -12 \][/tex]
Therefore, [tex]\( g(-2) = -12 \)[/tex].
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = -3\left(\frac{1}{2}\right)^{-1} = -3 \times 2 = -6 \][/tex]
Therefore, [tex]\( g(-1) = -6 \)[/tex].
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = -3 \left(\frac{1}{2}\right)^0 = -3 \times 1 = -3 \][/tex]
Therefore, [tex]\( g(0) = -3 \)[/tex].
4. For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = -3\left(\frac{1}{2}\right)^1 = -3 \times \frac{1}{2} = -1.5 \][/tex]
Therefore, [tex]\( g(1) = -1.5 \)[/tex].
So, the table with the corresponding values filled in is:
\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}
### Graphing the Points
The points to be plotted on the graph are:
- [tex]\((-2, -12)\)[/tex]
- [tex]\((-1, -6)\)[/tex]
- [tex]\((0, -3)\)[/tex]
- [tex]\((1, -1.5)\)[/tex]
On a coordinate plane:
- The x-axis should have values at least from [tex]\(-2\)[/tex] to [tex]\(1\)[/tex].
- The y-axis should accommodate values at least from [tex]\(-12\)[/tex] to [tex]\(0\)[/tex].
Once you plot these points, you will be able to visualize the function [tex]\( g(x) = -3\left(\frac{1}{2}\right)^x \)[/tex]. Ensure the points are accurately placed according to the values calculated.
