To graph the equation [tex]\( y = 3^x \)[/tex], we'll first fill in the table with the corresponding [tex]\( y \)[/tex] values for the given [tex]\( x \)[/tex] values.
Here is the table filled with the [tex]\( y \)[/tex] values:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $y$ \\
\hline
0 & 1 \\
\hline
1 & 3 \\
\hline
2 & 9 \\
\hline
\end{tabular}
\][/tex]
Now, let’s plot these points on a coordinate graph. The points are:
- [tex]\((0, 1)\)[/tex]
- [tex]\((1, 3)\)[/tex]
- [tex]\((2, 9)\)[/tex]
Here’s how you can graph these points:
1. Draw two perpendicular lines on the graph paper. Label the horizontal line as the [tex]\( x \)[/tex]-axis and the vertical line as the [tex]\( y \)[/tex]-axis.
2. Mark equal intervals on both the [tex]\( x \)[/tex]-axis and the [tex]\( y \)[/tex]-axis. Ensure the intervals are large enough to comfortably accommodate the points you need to plot.
3. Plot the point [tex]\((0, 1)\)[/tex]. Start at the origin (0, 0). Move 0 units right (since [tex]\( x = 0 \)[/tex]) and 1 unit up (since [tex]\( y = 1 \)[/tex]). Place a dot at this point.
4. Plot the point [tex]\((1, 3)\)[/tex]. Start at the origin. Move 1 unit to the right and 3 units up. Place a dot at this point.
5. Plot the point [tex]\((2, 9)\)[/tex]. Start at the origin. Move 2 units to the right and 9 units up. Place a dot at this point.
Now connect the points with a smooth curve, since the graph of [tex]\( y = 3^x \)[/tex] is an exponential curve.
Your graph should rise steeply to the right as [tex]\( x \)[/tex] increases, illustrating the exponential growth of the function.
