To determine the type of transformation that occurred when [tex]\( f(x) = (2x - 3)^3 \)[/tex] is transformed to [tex]\( g(x) = (-2x - 3)^3 \)[/tex], let's analyze the changes step-by-step.
1. Understanding f(x):
The original function is [tex]\( f(x) = (2x - 3)^3 \)[/tex].
- Here, the expression inside the parentheses is [tex]\( 2x - 3 \)[/tex].
2. Understanding g(x):
The transformed function is [tex]\( g(x) = (-2x - 3)^3 \)[/tex].
- Here, the expression inside the parentheses is [tex]\( -2x - 3 \)[/tex].
3. Identify the Changes:
- Compare the expressions inside the parentheses of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
- [tex]\( 2x - 3 \)[/tex] in [tex]\( f(x) \)[/tex]
- [tex]\( -2x - 3 \)[/tex] in [tex]\( g(x) \)[/tex]
- The coefficient of [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] is [tex]\( 2 \)[/tex], whereas, in [tex]\( g(x) \)[/tex], the coefficient of [tex]\( x \)[/tex] is [tex]\( -2 \)[/tex].
- The constant term [tex]\(-3\)[/tex] remains unchanged in both functions.
4. Effects of the Changes:
- Changing the coefficient of [tex]\( x \)[/tex] from [tex]\( 2 \)[/tex] to [tex]\( -2 \)[/tex] indicates a reflection over the y-axis.
- A horizontal reflection implies that each point [tex]\((x, y)\)[/tex] on the curve of [tex]\( f(x) \)[/tex] will map to the point [tex]\((-x, y)\)[/tex] on the curve of [tex]\( g(x) \)[/tex].
Based on the analysis above, the type of transformation that occurred is D. horizontal reflection.
