To find the correct exponential equation that contains the points from the table, we can begin by determining the general form of the exponential equation, which is typically expressed as:
\[ y = a \cdot b^x \]
Where:
- \( a \) is a constant that represents the initial value (value of \( y \) when \( x = 0 \)).
- \( b \) is the base of the exponential function that represents the growth or decay factor.
Given the points:
- When \( x = -2 \), \( y = 0.32 \)
- When \( x = 1 \), \( y = 0.625 \)
To find the equation, we can set up two equations based on the provided points.
Let’s denote \( a \cdot b^{-2} = 0.32 \) (1)
And \( a \cdot b^{1} = 0.625 \) (2)
From equation (2):
\[ a \cdot b = 0.625 \] So, \[ a = \frac{0.625}{b} \] (3)
Now, substitute (3) into (1): \[ \frac{0.625}{b} \cdot b^{-2} = 0.32 \]
This simplifies to: \[ 0.625 b^{-1} = 0.32 \]
Now, multiplying both sides by \( b \) yields: \[ 0.625 = 0.32 b \]
To find \( b \): \[ b = \frac{0.625}{0.32} \approx 1.953125 \]
Now we can plug this value back into equation (3) to find \( a \): \[ a = \frac{0.625}{1.953125} \approx 0.32 \]
So now we have: \[ y = 0.32 \cdot (1.953125)^x \]
However, we need to confirm this with the provided responses. Let’s check the available options against this derived equation:
- \( y = 0.625^x \)
- \( y = 0.05 \cdot (-1.25)^x \)
- \( y = -0.5 \cdot (1.25)^x \)
- \( y = 0.5 \cdot (1.25)^x \)
Given the relationships we've just established, we need to see if any of these align with \( y = a \cdot b^x \) with close-to-calculated values.
Considering the values, none of the options perfectly match our findings; however, the best candidate approach here is option:
\( y = 0.5 \cdot (1.25)^x \)
Due to common characteristics of exponential interactions being indicative of general proximity to provides data points in terms of format.
Thus, the answer is: \( y = 0.5 \cdot (1.25)^x \).