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

x
-2

-1

0

1

Now, graph the function.
Plot two points to graph the function.

3 answers

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

Table

| \( x \) | \( y = \left(\frac{1}{3}\right)^x \) | |---------|-------------------------------------| | -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 | \( \left(\frac{1}{3}\right)^{1} = \frac{1}{3} \) |

Completed Table

| \( x \) | \( y \) | |---------|---------| | -2 | 9 | | -1 | 3 | | 0 | 1 | | 1 | \(\frac{1}{3} \approx 0.33\) |

Points to Graph

Now, we can plot two points to graph the function:

  1. For \( x = -2 \): \( ( -2, 9) \)
  2. For \( x = -1 \): \( ( -1, 3) \)

Graphing the Function

  1. Plot the points:

    • Point 1: \( (-2, 9) \)
    • Point 2: \( (-1, 3) \)

  2. Draw the curve:

    • Using these points, you can now sketch or use graphing software to draw the curve for the function \( y = \left(\frac{1}{3}\right)^x \), which will show a decreasing exponential function starting from a high value as \( x \) approaches negative infinity and converging towards 0 as \( x \) increases.

Note

Since I can't directly draw the graph here, you can use graphing tools like Desmos or graphing calculators to visualize it properly along with the plotted points.

what would the 0 and 1 on the x axis equal to y

Based on the function \( y = \left(\frac{1}{3}\right)^x \), we can calculate the \( y \) values for \( x = 0 \) and \( x = 1 \):

  1. For \( x = 0 \): \[ y = \left(\frac{1}{3}\right)^0 = 1 \]

  2. For \( x = 1 \): \[ y = \left(\frac{1}{3}\right)^1 = \frac{1}{3} \approx 0.33 \]

Summary of Values

  • When \( x = 0 \), \( y = 1 \).
  • When \( x = 1 \), \( y \approx 0.33 \).

These values can be included in the previously completed table:

| \( x \) | \( y \) | |---------|---------| | -2 | 9 | | -1 | 3 | | 0 | 1 | | 1 | \(\frac{1}{3}\) (approximately 0.33) |

If you have further questions or need additional calculations, feel free to ask!