To analyze the function \( y = \left(\frac{1}{2}\right)^{x-1} - 10 \), we can start with identifying the asymptote and plotting points to graph the function.
Step 1: Identify the Asymptote
The function \( y = \left(\frac{1}{2}\right)^{x-1} - 10 \) approaches the horizontal line \( y = -10 \) as \( x \) increases, since the term \( \left(\frac{1}{2}\right)^{x-1} \) tends to zero.
Equation of the asymptote: \[ y = -10 \]
Step 2: Calculate Points
We will calculate at least five points where \( x \) is an integer.
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For \( x = 0 \): \[ y = \left(\frac{1}{2}\right)^{0-1} - 10 = \left(\frac{1}{2}\right)^{-1} - 10 = 2 - 10 = -8 \]
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For \( x = 1 \): \[ y = \left(\frac{1}{2}\right)^{1-1} - 10 = \left(\frac{1}{2}\right)^{0} - 10 = 1 - 10 = -9 \]
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For \( x = 2 \): \[ y = \left(\frac{1}{2}\right)^{2-1} - 10 = \left(\frac{1}{2}\right)^{1} - 10 = \frac{1}{2} - 10 = -9.5 \]
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For \( x = 3 \): \[ y = \left(\frac{1}{2}\right)^{3-1} - 10 = \left(\frac{1}{2}\right)^{2} - 10 = \frac{1}{4} - 10 = -9.75 \]
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For \( x = 4 \): \[ y = \left(\frac{1}{2}\right)^{4-1} - 10 = \left(\frac{1}{2}\right)^{3} - 10 = \frac{1}{8} - 10 = -9.875 \]
Points
Here are the points we've calculated:
- \( (0, -8) \)
- \( (1, -9) \)
- \( (2, -9.5) \)
- \( (3, -9.75) \)
- \( (4, -9.875) \)
Summary of Points
We have five calculated points:
- \( (0, -8) \)
- \( (1, -9) \)
- \( (2, -9.5) \)
- \( (3, -9.75) \)
- \( (4, -9.875) \)
Graphing
To plot these points:
- Draw the horizontal asymptote at \( y = -10 \).
- Mark the calculated points on the graph, noting that as \( x \) increases, \( y \) approaches closer to -10 but never actually reaches it.
This will give us a clear view of the behavior of the function.
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