To determine which exponential equation contains the given points in the input-output table, we can analyze the values provided in the table and the options given.
From the input-output table, we see:
- When \( x = -2 \), \( y = 0.32 \)
- When \( x = 1 \), \( y = 0.625 \)
We need to see which of the provided options satisfies these points.
Let's check the equations:
-
Option 1: \( y = -0.5 \cdot (1.25)^x \)
- For \( x = -2 \): \[ y = -0.5 \cdot (1.25)^{-2} = -0.5 \cdot \frac{1}{(1.25^2)} \approx -0.5 \cdot 0.64 \approx -0.32 \] (not valid since \( y \) cannot be negative)
- This option is not valid.
-
Option 2: \( y = (0.625)^x \)
- For \( x = -2 \): \[ y = (0.625)^{-2} = \frac{1}{(0.625^2)} \approx \frac{1}{0.390625} \approx 2.56 \] (not valid)
- This option is not valid.
-
Option 3: \( y = 0.05 \cdot -(1.25)^x \)
- For \( x = -2 \): \[ y = 0.05 \cdot -(1.25)^{-2} = 0.05 \cdot -\frac{1}{1.5625} \approx 0.05 \cdot -0.64 \approx -0.032 \] (not valid)
- This option is also not valid.
-
Option 4: \( y = 0.5 \cdot (1.25)^x \)
- For \( x = -2 \): \[ y = 0.5 \cdot (1.25)^{-2} = 0.5 \cdot \frac{1}{1.5625} \approx 0.5 \cdot 0.64 = 0.32 \] (valid for \( x = -2 \))
- For \( x = 1 \): \[ y = 0.5 \cdot (1.25)^1 = 0.5 \cdot 1.25 = 0.625 \] (valid for \( x = 1 \))
The correct exponential equation that contains the points shown in the input-output table is:
\( y = 0.5 \cdot (1.25)^x \).
1 answer