Let's analyze the transformation given in the equation:
\[ 3f(x+2)^3 = 3(x+2)^3 \]
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Horizontal Translation: The function \( f(x) = x^3 \) undergoes a transformation to \( f(x+2) \). This indicates a horizontal shift to the left by 2 units. Generally, \( f(x + c) \) translates the graph horizontally \( c \) units to the left if \( c \) is positive, and to the right if \( c \) is negative. Since \( c = 2 \) here, the translation is to the left.
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Vertical Stretch: The presence of \( 3f(x+2)^3 \) as opposed to just \( f(x) \) implies a vertical stretch. The factor multiplying \( f(x) \) (which is 3 in this case) indicates a vertical stretch by a factor of 3.
Putting it all together, we can state:
This translates the graph horizontally to the left and then stretched vertically by a factor of 3.