To find the exponential equation that fits the points given in the problem, we can use the general form of an exponential function:
\[ y = ab^x \]
where \( a \) is the initial value (when \( x = 0 \)) and \( b \) is the base that determines the rate of decay.
Given the points (-1, 10) and (1, 3.6), we can set up a system of equations to find \( a \) and \( b \).
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For the point (-1, 10): \[ 10 = ab^{-1} \quad \text{(Equation 1)} \] This simplifies to: \[ 10 = \frac{a}{b} \quad \Rightarrow \quad a = 10b \quad \text{(Equation 1.1)} \]
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For the point (1, 3.6): \[ 3.6 = ab^{1} \quad \text{(Equation 2)} \] This can be written as: \[ 3.6 = ab \quad \text{(Equation 2.1)} \]
Now, we can substitute Equation 1.1 into Equation 2.1:
\[ 3.6 = (10b)b \] \[ 3.6 = 10b^2 \]
Now, solve for \( b^2 \): \[ b^2 = \frac{3.6}{10} = 0.36 \] \[ b = \sqrt{0.36} = 0.6 \]
Now that we have found \( b \), we can use Equation 1.1 to find \( a \): \[ a = 10b = 10 \times 0.6 = 6 \]
So, the exponential equation is: \[ y = 6(0.6)^x \]
Thus, the exponential equation shown in the graph is:
\[ y = 6(0.6)^x \]
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