Use the drawing tool(s) to form the correct answers on the provided grid.

Consider the function [tex]g[/tex]:
[tex]\[ g(x) = 6 \left( \frac{3}{2} \right)^x \][/tex]

For the [tex]x[/tex]-values given in the table below, determine the corresponding values of [tex]g(x)[/tex] and plot each point on the graph.

[tex]\[
\begin{tabular}{|c|c|c|c|c|}
\hline
x & -1 & 0 & 1 & 2 \\
\hline
g(x) & & & & \\
\hline
\end{tabular}
\][/tex]



Answer :

To determine the corresponding values of [tex]\( g(x) \)[/tex] for the given [tex]\( x \)[/tex]-values, we need to evaluate the function [tex]\( g(x) = 6 \left( \frac{3}{2} \right)^x \)[/tex] at each of these [tex]\( x \)[/tex]-values.

1. For [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = 6 \left( \frac{3}{2} \right)^{-1} = 6 \left( \frac{2}{3} \right) = 4 \][/tex]

2. For [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = 6 \left( \frac{3}{2} \right)^0 = 6 \cdot 1 = 6 \][/tex]

3. For [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = 6 \left( \frac{3}{2} \right)^1 = 6 \cdot \frac{3}{2} = 9 \][/tex]

4. For [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = 6 \left( \frac{3}{2} \right)^2 = 6 \cdot \left( \frac{3}{2} \right)^2 = 6 \cdot \frac{9}{4} = 13.5 \][/tex]

Now, we can fill in the table with the corresponding [tex]\( g(x) \)[/tex] values:

[tex]\[ \begin{tabular}{|c|c|c|c|c|} \hline x & -1 & 0 & 1 & 2 \\ \hline g(x) & 4 & 6 & 9 & 13.5 \\ \hline \end{tabular} \][/tex]

Next, we need to plot these points on the graph.

The points to plot are:
- [tex]\( (-1, 4) \)[/tex]
- [tex]\( (0, 6) \)[/tex]
- [tex]\( (1, 9) \)[/tex]
- [tex]\( (2, 13.5) \)[/tex]

These points can be plotted on a [tex]\( (x, y) \)[/tex] coordinate plane. Make sure to label your axis and plot each point accurately.