To find the exponential equation of the form \(y = ab^x\), we can use the points given in the problem: (-1, 8) and (2, 0.125).
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Use the first point (-1, 8): \[ 8 = ab^{-1} \quad \Rightarrow \quad 8 = \frac{a}{b} \quad \Rightarrow \quad a = 8b \]
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Use the second point (2, 0.125): \[ 0.125 = ab^2 \] Now substitute \(a = 8b\) into this equation: \[ 0.125 = (8b)b^2 \quad \Rightarrow \quad 0.125 = 8b^3 \] Therefore, \[ b^3 = \frac{0.125}{8} = \frac{1}{64} \] Taking the cube root of both sides gives: \[ b = \frac{1}{4} \]
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Now substitute \(b\) back to find \(a\): \[ a = 8b = 8 \left(\frac{1}{4}\right) = 2 \]
So we have found \(a\) and \(b\): \[ a = 2 \quad \text{and} \quad b = \frac{1}{4} \]
- Write the final exponential equation: Combining \(a\) and \(b\) back into the exponential equation gives: \[ y = 2\left(\frac{1}{4}\right)^x \]
Thus, the exponential equation shown in the graph is: \[ y = 2 \left(\frac{1}{4}\right)^x \]
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