To model the given points \((0, 500)\), \((1, 450)\), and \((2, 405)\) with an exponential function, we can use the general form of an exponential function:
\[ y = a \cdot b^x \]
where:
- \( a \) is the initial value (the value when \( x = 0 \)),
- \( b \) is the base of the exponential (which determines the growth or decay rate),
- \( x \) is the independent variable,
- \( y \) is the dependent variable.
Given that the point \((0, 500)\) indicates that \( when , x = 0 \), \( y = 500\), we can directly state that:
\[ a = 500 \]
Thus, our equation now looks like:
\[ y = 500 \cdot b^x \]
Next, we can use the known points \((1, 450)\) and \((2, 405)\) to find the value of \( b \).
Using the point \((1, 450)\):
\[ 450 = 500 \cdot b^1 \]
This simplifies to:
\[ b = \frac{450}{500} = 0.9 \]
Now we can use the point \((2, 405)\) to validate our value of \( b \):
\[ 405 = 500 \cdot 0.9^2 \]
Calculating \( 0.9^2 \):
\[ 0.9^2 = 0.81 \]
Now substituting this back into the equation:
\[ 405 = 500 \cdot 0.81 = 405 \]
This confirms that our modeled data points fit perfectly into the exponential function with our calculated \( b \).
Thus, the final model is:
\[ y = 500 \cdot 0.9^x \]
Implementation in Python
Let's implement this function in Python to verify:
def exponential_model(x):
a = 500 # initial value
b = 0.9 # decay factor
return a * (b ** x)
# Test the function with the known points
print(exponential_model(0)) # should return approximately 500
print(exponential_model(1)) # should return approximately 450
print(exponential_model(2)) # should return approximately 405
This function will model the exponential decay based on the points provided. When you run this code, it should return the approximate expected values for \( y \).
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