To write the equation of an exponential function in the form \( f(x) = a(b)^x \) given the points (0, -2) and (1, -3), we can use the following steps:
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Determine the value of \( a \): The point (0, -2) tells us about the y-intercept of the function. Since \( f(0) = a(b)^0 = a \), we have: \[ a = -2. \]
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Now, substitute \( a \) into the form: So now we have \( f(x) = -2(b)^x \).
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Use the second point (1, -3) to find \( b \): Using the point (1, -3): \[ f(1) = -2(b)^1 = -3. \] This simplifies to: \[ -2b = -3. \] Dividing both sides by -2: \[ b = \frac{3}{2}. \]
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Write the final function: Now substitute \( a \) and \( b \) back into the equation: \[ f(x) = -2\left(\frac{3}{2}\right)^x. \]
Thus, the equation of the function is: \[ f(x) = -2\left(\frac{3}{2}\right)^x. \]
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