To find the equation of the exponential function in the form \( y = a(b)^x \) that passes through the points \((-1, 10)\) and \( (1, 25) \), we can follow the steps outlined below:
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Set up the system of equations: Using the points given, we can set up two equations based on the general form \( y = a(b)^x \).
For the point \((-1, 10)\): \[ 10 = a(b)^{-1} \quad \Rightarrow \quad 10 = \frac{a}{b} \quad \Rightarrow \quad a = 10b \]
For the point \((1, 25)\): \[ 25 = a(b)^{1} \quad \Rightarrow \quad 25 = ab \]
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Substitute for \(a\): Now substitute \( a = 10b \) into the equation \( 25 = ab \): \[ 25 = (10b)(b) = 10b^2 \]
This simplifies to: \[ b^2 = \frac{25}{10} = 2.5 \quad \Rightarrow \quad b = \sqrt{2.5} = \frac{5}{\sqrt{10}} = \frac{\sqrt{10}}{4} \]
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Calculate \( a \): Now that we have the value of \(b\), we can find \(a\): \[ a = 10b = 10 \cdot \sqrt{2.5} \]
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Put it all together: The function can now be written as: \[ y = a(b)^x = (10 \cdot \sqrt{2.5}) \left( \sqrt{2.5} \right)^x \]
Simplifying gives us: \[ y = 10 \cdot (2.5)^{\frac{x + 1}{2}} \]
Thus, the exponential function that passes through the points \((-1, 10)\) and \((1, 25)\) is given by:
\[ \boxed{y = 10 \cdot (2.5)^{x+1}} \]