Complete the table for the function y=(1/3)^x

x y
–2
–1
1
2

Now, graph the function.

Plot two points to graph the function.

1 answer

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:

  1. For \( x = -2 \), \( y = 9 \) → Point: (-2, 9)
  2. 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)