Let's start by completing the table for the function \( y = \left(\frac{1}{6}\right)^x \) using the given \( x \) values.
Completing the Table
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For \( x = -2 \): \[ y = \left(\frac{1}{6}\right)^{-2} = 6^2 = 36 \]
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For \( x = -1 \): \[ y = \left(\frac{1}{6}\right)^{-1} = 6^1 = 6 \]
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For \( x = 0 \): \[ y = \left(\frac{1}{6}\right)^0 = 1 \]
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For \( x = 1 \): \[ y = \left(\frac{1}{6}\right)^1 = \frac{1}{6} \approx 0.1667 \]
Completed Table
| \( x \) | \( y \) | |----------|----------| | -2 | 36 | | -1 | 6 | | 0 | 1 | | 1 | 0.1667 |
Graphing the Function
To graph the function, plot the points from the completed table. You can plot the following points on a coordinate grid:
- Point for \( (-2, 36) \)
- Point for \( (-1, 6) \)
- Point for \( (0, 1) \)
- Point for \( (1, 0.1667) \)
Note: Since I cannot graph directly here, I will describe how you can graph them:
- For each pair \( (x, y) \), place a dot at that coordinate on the Cartesian plane.
- Connect the points with a smooth curve, keeping in mind that the function \( y = \left(\frac{1}{6}\right)^x \) will asymptotically approach the x-axis for large \( x \) but never actually touches it (as it approaches zero).
This function will show an exponential decay as \( x \) increases, starting from a high value when \( x \) is negative, and decreasing towards zero as \( x \) becomes positive.