To complete the table for the function \( y = \left(\frac{1}{7}\right)^x \), we will calculate \( y \) for each value of \( x \).
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For \( x = -2 \): \[ y = \left(\frac{1}{7}\right)^{-2} = 7^2 = 49 \]
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For \( x = -1 \): \[ y = \left(\frac{1}{7}\right)^{-1} = 7^1 = 7 \]
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For \( x = 0 \): \[ y = \left(\frac{1}{7}\right)^{0} = 1 \]
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For \( x = 1 \): \[ y = \left(\frac{1}{7}\right)^{1} = \frac{1}{7} \]
Now we can fill in the table:
| \( x \) | \( y \) | |:---------:|:---------:| | -2 | 49 | | -1 | 7 | | 0 | 1 | | 1 | \(\frac{1}{7} \approx 0.14 \) |
Graphing the function:
To graph the function \( y = \left(\frac{1}{7}\right)^x \), we can plot the points calculated:
- For \( x = -2 \), \( y = 49 \)
- For \( x = -1 \), \( y = 7 \)
- For \( x = 0 \), \( y = 1 \)
- For \( x = 1 \), \( y \approx 0.14 \)
Here are two key points to graph:
- \( (-2, 49) \)
- \( (0, 1) \)
Graphing Points:
To graph, you would mark these points on a coordinate plane:
- The point \( (-2, 49) \) will be very high on the y-axis.
- The point \( (0, 1) \) will be on the y-axis.
- The point \( (1, \frac{1}{7}) \approx (1, 0.14) \) will be just above the x-axis.
The curve will decrease as x increases because the base \( \frac{1}{7} < 1 \).
If you were to create a visual:
- Draw the axes.
- Plot the points mentioned.
- Sketch a curve that approaches the x-axis as x increases and rises steeply as x decreases.
This function is an exponential decay function since the base is a fraction.
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