To derive the exponential equation of the form \( y = ab^x \) based on the points given, we'll use the two points \((-2, 1.28)\) and \( (1, 2.5) \).
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Set up equations using the two points.
For the point \((-2, 1.28)\): \[ 1.28 = ab^{-2} \]
For the point \( (1, 2.5) \): \[ 2.5 = ab^1 = ab \]
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Rearranging the equations: From the second equation, we can express \( a \): \[ a = \frac{2.5}{b} \]
Now, substitute \( a \) in the first equation: \[ 1.28 = \left(\frac{2.5}{b}\right) b^{-2} \] \[ 1.28 = \frac{2.5}{b^3} \]
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Solving for \( b \): Rearranging gives: \[ b^3 = \frac{2.5}{1.28} \] \[ b^3 \approx 1.953125 \] Taking the cube root: \[ b \approx \sqrt[3]{1.953125} \approx 1.26 \]
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Substituting back to find \( a \): Now substitute \( b \) back into \( a = \frac{2.5}{b} \): \[ a \approx \frac{2.5}{1.26} \approx 1.98 \]
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Write the final equation: Therefore, the exponential equation is: \[ y \approx 1.98 \cdot (1.26)^x \]
To create an accurate model, further adjustment with exact calculations might be necessary, but the estimated values yield: \[ y \approx 1.98 \cdot (1.26)^x \]
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