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 \( x \):
-
For \( x = -2 \): \[ y = 3^{1 - 2(-2)} = 3^{1 + 4} = 3^5 = 243 \]
-
For \( x = -1 \): \[ y = 3^{1 - 2(-1)} = 3^{1 + 2} = 3^3 = 27 \]
-
For \( x = 0 \): \[ y = 3^{1 - 2(0)} = 3^{1} = 3 \]
-
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:
- Create a coordinate system with an appropriate scale.
- Plot point \( (-2, 243) \) high on the y-axis.
- 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.