Let's analyze the transformations of the function [tex]\( g(x) = -(2)^{x+4} - 2 \)[/tex] step by step, comparing it to the parent function [tex]\( f(x) = 2^x \)[/tex].
1. Horizontal Shift:
- The expression inside the exponent, [tex]\( x+4 \)[/tex], indicates a horizontal shift. Specifically, it represents a shift of the graph to the left by 4 units. This is because adding 4 inside the function moves the graph in the negative [tex]\( x \)[/tex]-direction.
2. Reflection:
- The negative sign outside the exponential term, [tex]\(-(2)^{x+4}\)[/tex], indicates a reflection. A negative sign before the entire function reflects the graph across the [tex]\( x \)[/tex]-axis.
3. Vertical Shift:
- The constant term outside the exponential, [tex]\(-2\)[/tex], indicates a vertical shift. Since this term is subtracted, it shifts the graph downward by 2 units.
Summarizing these transformations:
- There is a horizontal shift to the left by 4 units.
- The graph is reflected over the [tex]\( x \)[/tex]-axis.
- The graph is shifted down by 2 units.
Thus, the correct description of the transformations from the parent function [tex]\( f(x) = 2^x \)[/tex] to the transformed function [tex]\( g(x) = -(2)^{x+4} - 2 \)[/tex] is:
shift 4 units left, reflect over the [tex]\( x \)[/tex]-axis, shift 2 units down.
