Question

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

Answer 1

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:

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