To find the exponential equation that contains the points given in the input-output table, we will analyze the data points:
- When \( x = -2 \), \( y = 5,000,000 \).
- When \( x = 1 \), \( y = 0.625 \).
Let's check each of the provided response options:
Option 1: \( y = 125 \cdot (-0.005)^x \)
-
For \( x = -2 \): \[ y = 125 \cdot (-0.005)^{-2} = 125 \cdot \frac{1}{(-0.005)^2} = 125 \cdot 40000 = 5000000 \] (This is correct)
-
For \( x = 1 \): \[ y = 125 \cdot (-0.005)^1 = 125 \cdot (-0.005) = -0.625 \] (This is incorrect as y should be positive)
Option 2: \( y = -125 \cdot (0.005)^x \)
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For \( x = -2 \): \[ y = -125 \cdot (0.005)^{-2} = -125 \cdot 40000 = -5000000 \] (This is incorrect)
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For \( x = 1 \): \[ y = -125 \cdot (0.005)^1 = -125 \cdot 0.005 = -0.625 \] (Again, this is incorrect)
Option 3: \( y = 125 \cdot (0.005)^x \)
-
For \( x = -2 \): \[ y = 125 \cdot (0.005)^{-2} = 125 \cdot 40000 = 5000000 \] (This is correct)
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For \( x = 1 \): \[ y = 125 \cdot (0.005)^1 = 125 \cdot 0.005 = 0.625 \] (This is also correct)
Option 4: \( y = 0.0625^x \)
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For \( x = -2 \): \[ y = 0.0625^{-2} = 16 \] (This is incorrect)
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For \( x = 1 \): \[ y = 0.0625^1 = 0.0625 \] (This is also incorrect)
Conclusion:
The correct exponential equation that contains the points shown in the table is:
\[ \boxed{y = 125 \cdot (0.005)^x} \]