Question

Complete the table for the function y= 3(1/9)^x.
x y
-2 __
-1 __
0 __
1 __
Now, graph the function.
Plot two points to graph the function

Answer 1

To complete the table for the function \( y = 3\left(\frac{1}{9}\right)^x \), we will calculate \( y \) for each given value of \( x \).

First, let's express \( \frac{1}{9} \) in terms of powers of 3:
\[
\frac{1}{9} = 3^{-2}
\]
Thus, we can rewrite the function as:
\[
y = 3(3^{-2})^x = 3 \cdot 3^{-2x}
\]
This simplifies to:
\[
y = 3^{1 - 2x}
\]

### Now, let's calculate the values for each \( x \):

1. **For \( x = -2 \)**:
\[
y = 3^{1 - 2(-2)} = 3^{1 + 4} = 3^5 = 243
\]

2. **For \( x = -1 \)**:
\[
y = 3^{1 - 2(-1)} = 3^{1 + 2} = 3^3 = 27
\]

3. **For \( x = 0 \)**:
\[
y = 3^{1 - 2(0)} = 3^{1} = 3
\]

4. **For \( x = 1 \)**:
\[
y = 3^{1 - 2(1)} = 3^{1 - 2} = 3^{-1} = \frac{1}{3}
\]

### The completed table will look like this:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 243 \\
-1 & 27 \\
0 & 3 \\
1 & \frac{1}{3} \\
\hline
\end{array}
\]

### Now, let's plot the two points:

For graphing, we can use the first two calculated points that are significant in demonstrating the function's behavior:
- \( (-2, 243) \)
- \( (-1, 27) \)

### Note:
Since I cannot provide a visual graph, here's how you would plot:

1. Create a coordinate system with an appropriate scale.
2. Plot point \( (-2, 243) \) high on the y-axis.
3. Plot point \( (-1, 27) \) lower than the first but still notably high.

This function will show an exponential decay as \( x \) increases, approaching but never quite reaching \( y = 0 \) as \( x \) becomes larger. The points will help illustrate this rapid decrease.

If you have graph paper or a graphing tool, you can draw the curve based on these values. The points will help provide a visual reference for the function behavior.