Question

Use the table to answer the question.

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

x y
−2
−
2
0.32
0.32

1 0.625
0.625

(1 point)
Responses

y=(0.625)x
𝑦
=
(
0.625
)
𝑥
y is equal to 0 point 6 2 5 to the x th power

y=−0.5⋅(1.25)x
𝑦
=
−
0.5
⋅
(
1.25
)
𝑥
y is equal to negative 0 point 5 times 1 point 2 5 to the x th power

y=0.5⋅(1.25)x
𝑦
=
0.5
⋅
(
1.25
)
𝑥
y is equal to 0 point 5 times 1 point 2 5 to the x th power

y=0.05⋅−(1.25)x

Answer 1

To determine which exponential equation is consistent with the given points from the table, we need to find an equation that matches the values of \(y\) for corresponding \(x\) values.

From the table:
- For \(x = -2\), \(y = 0.32\)
- For \(x = 1\), \(y = 0.625\)

Let's evaluate the given options one by one:

1. **Option**: \(y = (0.625)^x\)

For \(x = -2\):
\[
y = (0.625)^{-2} = \frac{1}{(0.625)^2} \approx \frac{1}{0.390625} \approx 2.56 \quad (\text{not } 0.32)
\]

2. **Option**: \(y = -0.5 \cdot (1.25)^x\)

For \(x = -2\):
\[
y = -0.5 \cdot (1.25)^{-2} = -0.5 \cdot \frac{1}{(1.25)^2} = -0.5 \cdot \frac{1}{1.5625} \approx -0.32 \quad (\text{not } 0.32)
\]

3. **Option**: \(y = 0.5 \cdot (1.25)^x\)

For \(x = -2\):
\[
y = 0.5 \cdot (1.25)^{-2} = 0.5 \cdot \frac{1}{(1.25)^2} = 0.5 \cdot \frac{1}{1.5625} \approx 0.32 \quad (\text{this matches})
\]

Now checking \(x = 1\):
\[
y = 0.5 \cdot (1.25)^1 = 0.5 \cdot 1.25 = 0.625 \quad (\text{this matches too})
\]

4. **Option**: \(y = 0.05 \cdot -(1.25)^x\)

For \(x = -2\):
\[
y = 0.05 \cdot -(1.25)^{-2} = 0.05 \cdot -\frac{1}{(1.25)^2} = 0.05 \cdot -\frac{1}{1.5625} \approx -0.032 \quad (\text{not } 0.32)
\]

The only equation that produces the correct outputs for both given inputs is:

**Final answer: \(y = 0.5 \cdot (1.25)^x\)**.