Use the table to answer the question.

Question

Which exponential equation contains the points shown in the input-output table?

x y
−2
−
2
5,000,000
1 0.625
(1 point)
Responses

y=0.0625x
𝑦
=
0.0625
𝑥
y is equal to 0 point 0 6 2 5 to the x th power

y=125⋅−0.005x
𝑦
=
125
⋅
−
0.005
𝑥
y is equal to 125 times negative 0 point 0 0 5 to the x th power

y=−125⋅0.005x
𝑦
=
−
125
⋅
0.005
𝑥
y is equal to negative 125 times 0 point 0 0 5 to the x th power

y=125⋅0.005x

Answer 1

To determine which exponential equation contains the points from the input-output table, we need to analyze the values of x and y provided.

We have:

• For \( x = -2 \), \( y = 5,000,000 \)
• For \( x = 1 \), \( y = 0.625 \)

Now let’s evaluate each option:

1. Option 1: \( y = 0.0625^x \)

   • For \( x = -2 \): \( y = 0.0625^{-2} = \frac{1}{0.0625^2} = \frac{1}{0.00390625} = 256 \) (not equal to 5,000,000)
   • For \( x = 1 \): \( y = 0.0625^{1} = 0.0625 \) (not equal to 0.625)

2. Option 2: \( y = 125 \cdot (-0.005)^x \)

   • For \( x = -2 \): \( y = 125 \cdot (-0.005)^{-2} = 125 \cdot 40000 = 5,000,000 \) (matches)
   • For \( x = 1 \): \( y = 125 \cdot (-0.005)^{1} = 125 \cdot -0.005 = -0.625 \) (not equal to 0.625)

3. Option 3: \( y = -125 \cdot 0.005^x \)

   • For \( x = -2 \): \( y = -125 \cdot (0.005)^{-2} = -125 \cdot 40000 = -5,000,000 \) (not equal to 5,000,000)
   • For \( x = 1 \): \( y = -125 \cdot 0.005 = -0.625 \) (not equal to 0.625)

4. Option 4: \( y = 125 \cdot 0.005^x \)

   • For \( x = -2 \): \( y = 125 \cdot (0.005)^{-2} = 125 \cdot 40000 = 5,000,000 \) (matches)
   • For \( x = 1 \): \( y = 125 \cdot 0.005 = 0.625 \) (matches)

Conclusion:

The only candidate that matches both points (5,000,000 for \( x = -2 \) and 0.625 for \( x = 1 \)) is: \( y = 125 \cdot 0.005^x \).