To determine 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. Identify Horizontal Translation:
The expression inside the cube, [tex]\((x + 2)\)[/tex], indicates a horizontal shift. The function [tex]\( (x + 2) \)[/tex] means that [tex]\( f(x) \)[/tex] is shifted to the left by 2 units. Therefore, the function [tex]\( f \)[/tex] is translated 2 units to the left.
2. Identify Vertical Translation:
The [tex]\(-4\)[/tex] at the end of the expression indicates a vertical shift. A subtraction outside the function moves it down. Therefore, [tex]\( f \)[/tex] is translated 4 units down.
3. Identify Reflection:
The negative sign in front of the entire function, [tex]\(-(x+2)^3\)[/tex], indicates a reflection. A negative sign outside the function reflects the function across the x-axis. Therefore, [tex]\( f \)[/tex] is reflected across the x-axis.
Putting it all together:
- The function [tex]\( f \)[/tex] is translated 2 units to the left.
- The function [tex]\( f \)[/tex] is translated 4 units down.
- The function [tex]\( f \)[/tex] is reflected across the x-axis.
So, the correct selections are:
- 2 units [tex]\(\boxed{\text{left}}\)[/tex]
- 4 units [tex]\(\boxed{\text{down}}\)[/tex]
- reflected across the [tex]\(\boxed{\text{x-axis}}\)[/tex]
