To create the function [tex]\( h(x) = -(x+2)^3 - 4 \)[/tex] from the original function [tex]\( f(x) = x^3 \)[/tex], we need to identify the transformations step-by-step:
1. Horizontal Translation:
The term [tex]\((x + 2)\)[/tex] inside the function indicates a horizontal shift. Since the value [tex]\(+2\)[/tex] appears inside the parentheses, it translates the graph to the left by 2 units (remember that adding a positive number inside the function results in a shift to the left, while adding a negative number would result in a shift to the right).
Therefore, the horizontal translation is 2 units to the left.
2. Reflection:
The negative sign in front of the function [tex]\(-(x + 2)^3\)[/tex] indicates a reflection. Since the negative sign affects the entire function, the reflection is across the x-axis.
Therefore, the reflection is across the x-axis.
3. Vertical Translation:
The [tex]\(-4\)[/tex] at the end of the expression [tex]\(-(x + 2)^3 - 4\)[/tex] indicates a vertical shift. Since it's a negative value, it translates the graph downwards by 4 units.
Therefore, the vertical translation is 4 units down.
So, putting it all together, function [tex]\( f(x) = x^3 \)[/tex] was transformed as follows to create function [tex]\( h(x) = -(x+2)^3 - 4 \)[/tex]:
- Translated 2 units to the left.
- Translated 4 units down.
- Reflected across the x-axis.
