To find the transformations applied to the function [tex]\( f(x) = x^3 \)[/tex] to obtain [tex]\( h(x) = -(x+2)^3 - 4 \)[/tex], follow these steps:
1. Observe the inside of the cubic function [tex]\( -(x+2)^3 \)[/tex]. The [tex]\( +2 \)[/tex] inside the parenthesis indicates a horizontal translation. Since it's [tex]\( +2 \)[/tex], the translation is 2 units to the left.
2. Next, observe the [tex]\( -4 \)[/tex] outside of the cubic function [tex]\( (x+2)^3 \)[/tex]. This represents a vertical translation. Since it is [tex]\( -4 \)[/tex], the function is translated 4 units down.
3. Lastly, look at the negative sign in front of the cubic function [tex]\( -(x+2)^3 \)[/tex]. This negative sign indicates a reflection. Specifically, it's a reflection across the x-axis.
So, the transformations are:
1. The function was translated 2 units to the left.
2. The function was translated 4 units down.
3. The function was reflected across the x-axis.
Therefore, the correct answers for the drop-down menus are:
- Translated 2 units left.
- Translated 4 units down.
- Reflected across the x-axis.
