To complete the table for the function \( y = 0.1^x \), we can substitute each value of \( x \) to find the corresponding \( y \) values.
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For \( x = -2 \): \[ y = 0.1^{-2} = \frac{1}{0.1^2} = \frac{1}{0.01} = 100 \]
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For \( x = -1 \): \[ y = 0.1^{-1} = \frac{1}{0.1} = 10 \]
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For \( x = 0 \): \[ y = 0.1^{0} = 1 \]
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For \( x = 1 \): \[ y = 0.1^{1} = 0.1 \]
Now, we can fill out the table:
| x | y | |-----|--------| | -2 | 100 | | -1 | 10 | | 0 | 1 | | 1 | 0.1 |
Graphing the Function
To graph the function \( y = 0.1^x \), we can plot the points we calculated:
- Point for \( ( -2, 100 ) \)
- Point for \( ( -1, 10 ) \)
- Point for \( ( 0, 1 ) \)
- Point for \( ( 1, 0.1 ) \)
Let's plot two of these points:
- Point 1: \( (-2, 100) \)
- Point 2: \( (0, 1) \)
It's important to note that as \( x \) increases, \( y \) decreases, tending toward zero but never actually reaching it. The graph of \( y = 0.1^x \) will decrease rapidly for positive values of \( x \) and increase sharply for negative values of \( x \).
If you draw this on a coordinate grid, you would see that the curve approaches the x-axis but does not touch it, illustrating the exponential decay of the function.