To find the transformed function after applying a vertical compression by a factor of \( \frac{1}{2} \) and then translating the equation 3 units to the right, we can follow these steps:
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Vertical Compression: The original function is: \[ F(x) = 6(x - 2)^2 \] To compress it vertically by a factor of \( \frac{1}{2} \), we multiply the entire function by \( \frac{1}{2} \): \[ G(x) = \frac{1}{2} \cdot 6(x - 2)^2 = 3(x - 2)^2 \]
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Translation: To translate the function 3 units to the right, we need to replace \( x \) with \( (x - 3) \) in the function \( G(x) \): \[ H(x) = G(x - 3) = 3((x - 3) - 2)^2 \] Simplifying the expression inside the parentheses: \[ H(x) = 3((x - 5)^2) \]
Thus, after vertically compressing by a factor of \( \frac{1}{2} \) and translating 3 units to the right, the transformed function is: \[ H(x) = 3(x - 5)^2 \]