To solve this problem, we'll follow these steps:
1. Identify the given x-values: The x-values provided are -2, -1, 0, and 1.
2. Determine the corresponding values of [tex]\( g(x) \)[/tex]: We need to find the values of [tex]\( g(x) \)[/tex] using the function [tex]\( g(x) = -3 \left(\frac{1}{2}\right)^x \)[/tex] for each x-value.
3. Fill in the table with the computed values.
4. Plot each point on the graph using the function [tex]\( g(x) = -3 \left(\frac{1}{2}\right)^x \)[/tex].
Let's start with the calculations:
1. For [tex]\( x = -2 \)[/tex]:
[tex]\[
g(-2) = -3 \left(\frac{1}{2}\right)^{-2} = -3 \left(2^2\right) = -3 \times 4 = -12.0
\][/tex]
2. For [tex]\( x = -1 \)[/tex]:
[tex]\[
g(-1) = -3 \left(\frac{1}{2}\right)^{-1} = -3 \left(2\right) = -3 \times 2 = -6.0
\][/tex]
3. For [tex]\( x = 0 \)[/tex]:
[tex]\[
g(0) = -3 \left(\frac{1}{2}\right)^{0} = -3 \times 1 = -3.0
\][/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]
Now, let's fill in the table:
\begin{tabular}{|c|c|c|c|c|}
\hline
[tex]$x$[/tex] & -2 & -1 & 0 & 1 \\
\hline
[tex]$g(x)$[/tex] & -12.0 & -6.0 & -3.0 & -1.5 \\
\hline
\end{tabular}
Next, let's plot these points on the graph. Here are the coordinates we need to plot:
- (-2, -12.0)
- (-1, -6.0)
- (0, -3.0)
- (1, -1.5)
Use the provided grid to mark these points accurately:
1. Start at the origin (0, 0).
2. Move left 2 units and down 12 units to plot (-2, -12.0).
3. Move left 1 unit and down 6 units to plot (-1, -6.0).
4. Remain at the origin and move down 3 units to plot (0, -3.0).
5. Move right 1 unit and down 1.5 units to plot (1, -1.5).
Once all points are plotted, you can connect them smoothly to visualize the behavior of the function [tex]\( g(x) = -3 \left(\frac{1}{2}\right)^x \)[/tex].
