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Use the table to answer the question. Which exponential equation contains the points shown in the input-output table? x y −2 5,000,000 1 0.625 (1 point) Responses y=125⋅−0.005x y is equal to 125 times negative 0 point 0 0 5 to the x th power y=−125⋅0.005x y is equal to negative 125 times 0 point 0 0 5 to the x th power y=125⋅0.005x y is equal to 125 times 0 point 0 0 5 to the x th power y=0.0625x

1 answer

To find the exponential equation that contains the points given in the input-output table, we will analyze the data points:

  1. When \( x = -2 \), \( y = 5,000,000 \).
  2. 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: \( y = -125 \cdot (0.005)^x \)

  • For \( x = -2 \): \[ y = -125 \cdot (0.005)^{-2} = -125 \cdot 40000 = -5000000 \] (This is incorrect)

  • 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)

  • For \( x = 1 \): \[ y = 125 \cdot (0.005)^1 = 125 \cdot 0.005 = 0.625 \] (This is also correct)

Option 4: \( y = 0.0625^x \)

  • For \( x = -2 \): \[ y = 0.0625^{-2} = 16 \] (This is incorrect)

  • 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:

\[ \boxed{y = 125 \cdot (0.005)^x} \]