To describe the transformation from the parent cubic function [tex]\( y = x^3 \)[/tex] to the function [tex]\( y = -x + 3 \)[/tex], we need to analyze the changes applied to the graph of the original function.
1. Reflection over the y-axis: In the original cubic function [tex]\( y = x^3 \)[/tex], we get a reflection over the y-axis when we negate the variable [tex]\( x \)[/tex]. Thus, if we consider [tex]\( y = (-x)^3 \)[/tex], it would still be [tex]\( y = -x^3 \)[/tex]. In the given function [tex]\( y = -x + 3 \)[/tex], the presence of [tex]\( -x \)[/tex] indicates a reflection over the y-axis.
2. Vertical Shift: Next, let's look at the constant term in the transformed function [tex]\( y = -x + 3 \)[/tex]. This [tex]\( +3 \)[/tex] indicates that the entire graph of the function has been shifted vertically. A positive constant added to the function implies an upward shift. Therefore, the graph is shifted up by 3 units.
Based on this analysis, we can select the correct transformations as follows:
- Reflection over the y-axis
- Shift up 3 units
Thus, the correct options are:
1. Reflection over the y-axis
2. Shift up 3 units
