The order of the transformations undergone for f(x)=x^3 to 3f(x+2)^3=3(x+2)^3 is translated horizontally to the left and then stretched vertically

Question

True or False

Answer 1

To analyze the transformations from \( f(x) = x^3 \) to \( 3f(x+2)^3 = 3(x+2)^3 \), let's break it down step by step.

1. Starting Function: \( f(x) = x^3 \)

2. Horizontal Translation: The function is translated to the left by 2 units, which is indicated by the term \( f(x+2) \). This means: \[ f(x+2) = (x+2)^3 \]

3. Vertical Stretch: The expression \( 3f(x+2) \) indicates a vertical stretch by a factor of 3: \[ 3f(x+2) = 3(x+2)^3 \]

So, the transformation sequence is first a horizontal translation of 2 units to the left, followed by a vertical stretch by a factor of 3.

In contrast to the statement, if we interpret the statement as indicating the order of transformations ("translated horizontally to the left and then stretched vertically"), it correctly reflects the actual sequence of transformations.

Therefore, the statement is True.