To complete the table for the function \( y = \left( \frac{1}{3} \right)^x \), we can substitute the values of \( x \) into the function to find the corresponding \( y \) values.
Table of Values
\[ \begin{array}{|c|c|} \hline x & y \ \hline -2 & \left( \frac{1}{3} \right)^{-2} = 3^2 = 9 \ -1 & \left( \frac{1}{3} \right)^{-1} = 3^1 = 3 \ 0 & \left( \frac{1}{3} \right)^{0} = 1 \ 1 & \left( \frac{1}{3} \right)^{1} = \frac{1}{3} \ 2 & \left( \frac{1}{3} \right)^{2} = \frac{1}{9} \ \hline \end{array} \]
Completed Table
\[ \begin{array}{|c|c|} \hline x & y \ \hline -2 & 9 \ -1 & 3 \ 0 & 1 \ 1 & \frac{1}{3} \ 2 & \frac{1}{9} \ \hline \end{array} \]
Points to Plot
To graph the function, we can plot two points based on the completed table:
- For \( x = -2 \), \( y = 9 \) → Point: (-2, 9)
- For \( x = 1 \), \( y = \frac{1}{3} \approx 0.33 \) → Point: (1, 0.33)
Graph Description
- The graph of \( y = \left( \frac{1}{3} \right)^x \) is an exponential decay function.
- As \( x \) increases, \( y \) decreases towards zero but never actually reaches it.
- When \( x < 0 \), \( y \) increases rapidly.
Graph Visualization
You can sketch the graph by plotting the points (-2, 9) and (1, 0.33), and drawing a curve that approaches the x-axis (but never touches it) as \( x \) becomes positive.
To summarize the points for plotting:
- Point 1: (-2, 9)
- Point 2: (1, 0.33)