To find the exponential equation represented by the points (2, 5,000,000) and (1, 0.625), we can use the general form of an exponential function, which is:
\[ y = ab^x \]
where \( a \) is the initial value (the value when \( x = 0 \)), \( b \) is the base (growth or decay factor), and \( x \) is the exponent.
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Use the first point (2, 5,000,000): \[ 5,000,000 = ab^2 \]
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Use the second point (1, 0.625): \[ 0.625 = ab^1 \] Simplifying this gives: \[ 0.625 = ab \]
Now we have two equations:
- \( 5,000,000 = ab^2 \) (Equation 1)
- \( 0.625 = ab \) (Equation 2)
Next, we can solve these equations simultaneously. From Equation 2, solve for \( a \): \[ a = \frac{0.625}{b} \]
Substitute this expression for \( a \) into Equation 1: \[ 5,000,000 = \left(\frac{0.625}{b}\right) b^2 \] This simplifies to: \[ 5,000,000 = 0.625b \]
Now, solve for \( b \): \[ b = \frac{5,000,000}{0.625} = 8,000,000 \]
Now that we have \( b \), we can find \( a \). Substitute \( b \) back into Equation 2: \[ 0.625 = a(8,000,000) \] Thus, \[ a = \frac{0.625}{8,000,000} = 7.8125 \times 10^{-8} \]
Now we have the values of \( a \) and \( b \):
- \( a \approx 7.8125 \times 10^{-8} \)
- \( b = 8,000,000 \)
Finally, we can put these into the exponential function: \[ y = (7.8125 \times 10^{-8})(8,000,000)^x \]
This is the exponential equation represented by the graph through points (2, 5,000,000) and (1, 0.625).
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