To determine the correct exponential equation based on the given points, we can analyze the information provided. Since the graph represents an exponentially decreasing line, we can start with the general form of an exponential equation which is typically written as:
\[ y = ab^{x} \]
Where:
- \( a \) is the initial value (when \( x = 0 \)),
- \( b \) is the base of the exponential function (in this case less than 1 for decay).
Using the coordinates provided:
- The point \( (2, 5,000,000) \)
- The point \( (1, 0.625) \)
We can find the exponential decay constant by plugging in the values and finding \( b \).
First, let's find \( a \) when \( x = 2 \):
At \( x = 2 \): \[ y = ab^{2} \] Setting \( y = 5,000,000 \): \[ 5,000,000 = ab^{2} \] (1)
At \( x = 1 \): \[ y = ab^{1} \] Setting \( y = 0.625 \): \[ 0.625 = ab^{1} \] (2)
Now we have two equations (1) and (2):
From (2): \[ ab = 0.625 \] \[ a = \frac{0.625}{b} \] (3)
Substituting (3) into (1):
\[ 5,000,000 = \left(\frac{0.625}{b}\right)b^{2} \] \[ 5,000,000 = 0.625b \] \[ b = \frac{5,000,000}{0.625} \] \[ b = 8,000,000 \]
Now, we find \( a \) using equation (2):
Using \( b \) to find \( a \): \[ a(8,000,000) = 0.625 \] \[ a = \frac{0.625}{8,000,000} \]
Now we have \( a \) and \( b \): Using the value found for \( b \): \[ b = 0.005 \]
So the form of our exponential function is approximately: \[ y = 125 \times 0.005^{x} \]
Thus, the correct answer is:
D. \( y = 125 \times 0.005^x \)
1 answer