To graph the function [tex]\( y = 3(2)^x \)[/tex], we need to populate the table with corresponding [tex]\(y\)[/tex]-values for the given [tex]\(x\)[/tex]-values. We will evaluate the function for [tex]\( x = 0 \)[/tex], [tex]\( x = 1 \)[/tex], and [tex]\( x = 2 \)[/tex].
First, let's plug in [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ y = 3(2)^0 \][/tex]
Since [tex]\( 2^0 = 1 \)[/tex]:
[tex]\[ y = 3 \times 1 = 3 \][/tex]
Therefore, when [tex]\( x = 0 \)[/tex], [tex]\( y = 3 \)[/tex].
Next, let's evaluate the function for [tex]\( x = 1 \)[/tex]:
[tex]\[ y = 3(2)^1 \][/tex]
Since [tex]\( 2^1 = 2 \)[/tex]:
[tex]\[ y = 3 \times 2 = 6 \][/tex]
Thus, when [tex]\( x = 1 \)[/tex], [tex]\( y = 6 \)[/tex].
Finally, for [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 3(2)^2 \][/tex]
Since [tex]\( 2^2 = 4 \)[/tex]:
[tex]\[ y = 3 \times 4 = 12 \][/tex]
So, for [tex]\( x = 2 \)[/tex], [tex]\( y = 12 \)[/tex].
Now, we can fill in the table with these values:
[tex]\[
\begin{tabular}{|c|l|}
\hline
$x$ & $y$ \\
\hline
0 & 3 \\
\hline
1 & 6 \\
\hline
2 & 12 \\
\hline
\end{tabular}
\][/tex]
With the table completed, we can plot these points [tex]\((0, 3)\)[/tex], [tex]\((1, 6)\)[/tex], and [tex]\((2, 12)\)[/tex] on a coordinate plane to graph the function [tex]\( y = 3(2)^x \)[/tex]. These points show an exponential growth pattern, and you can draw a smooth curve passing through these points to represent the function.
