To complete the table for the function \( y = 5(0.1)^x \), we will calculate the value of \( y \) for each given \( x \).
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For \( x = -2 \): \[ y = 5(0.1)^{-2} = 5 \cdot (10^2) = 5 \cdot 100 = 500 \]
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For \( x = -1 \): \[ y = 5(0.1)^{-1} = 5 \cdot (10^1) = 5 \cdot 10 = 50 \]
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For \( x = 0 \): \[ y = 5(0.1)^{0} = 5 \cdot 1 = 5 \]
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For \( x = 1 \): \[ y = 5(0.1)^{1} = 5 \cdot 0.1 = 0.5 \]
Now, we can fill in the table:
| x | y | |-----|-------| | -2 | 500 | | -1 | 50 | | 0 | 5 | | 1 | 0.5 |
Graphing the Function
To graph the function, you only need two points. Let's use the points we calculated:
- Point for \( x = -2 \): \( (-2, 500) \)
- Point for \( x = 0 \): \( (0, 5) \)
When graphing these points, you can use an x-y coordinate system where the x-axis represents the values of \( x \) and the y-axis represents the values of \( y \).
- The point \( (-2, 500) \) will be very high on the graph, and may not fit if you are using a standard scale.
- The point \( (0, 5) \) will be on the vertical line at \( x = 0 \) and up to 5 on the y-axis.
Since the function \( y = 5(0.1)^x \) decreases rapidly as \( x \) increases, you will see that for larger values of \( x \), the \( y \) values will approach zero.
If you were to graph more points, you could plot points for \( x = -1 \) and \( x = 1 \) as well:
- Point for \( x = -1 \): \( (-1, 50) \)
- Point for \( x = 1 \): \( (1, 0.5) \)
This will give you a clearer curve for the function, showing the exponential decay.