To find an exponential function of the form \( y = a(b)^x \) that fits the given points, we can first look for the values of \( a \) and \( b \).
Given the table of values:
- \( (0, 80) \)
- \( (1, 20) \)
- \( (2, 5) \)
- \( (3, 1.25) \)
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Identify the value of \( a \): The value of \( a \) is the \( y \)-value when \( x = 0 \): \[ a = 80 \]
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Determine the value of \( b \): We can use the ratio of consecutive \( y \)-values to find \( b \):
From \( x = 0 \) to \( x = 1 \): \[ \frac{y(1)}{y(0)} = \frac{20}{80} = \frac{1}{4} \]
From \( x = 1 \) to \( x = 2 \): \[ \frac{y(2)}{y(1)} = \frac{5}{20} = \frac{1}{4} \]
From \( x = 2 \) to \( x = 3 \): \[ \frac{y(3)}{y(2)} = \frac{1.25}{5} = \frac{1.25}{5} = \frac{1}{4} \]
Since the ratio of consecutive \( y \)-values is consistently \( \frac{1}{4} \), we can determine that: \[ b = \frac{1}{4} \]
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Write the function: Using the values of \( a \) and \( b \) we found, we can now write our exponential function: \[ y = 80\left( \frac{1}{4} \right)^x \]
Thus, the exponential function that models the points in the table is: \[ y = 80 \left( \frac{1}{4} \right)^x \]
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