To determine the transformations that have been applied to the parent function [tex]\( f(x) = 2^x \)[/tex] to obtain the function [tex]\( g(x) = -(2)^{x+4} - 2 \)[/tex], we need to analyze the changes step by step.
1. Horizontal Shift:
- The expression [tex]\( x + 4 \)[/tex] indicates a horizontal shift. Specifically, adding 4 inside the function argument (i.e., as [tex]\( x + 4 \)[/tex]) shifts the graph to the left by 4 units.
2. Reflection:
- The negative sign in front of the function [tex]\( -(2)^{x+4} \)[/tex] denotes a reflection. Since the negative sign is outside the exponent, it reflects the graph over the [tex]\( x \)[/tex]-axis.
3. Vertical Shift:
- The term [tex]\( -2 \)[/tex] at the end of the function [tex]\( -(2)^{x+4} - 2 \)[/tex] indicates a vertical shift. Subtracting 2 from the function shifts the graph down by 2 units.
Taking all these transformations into account, [tex]\( g(x) = -(2)^{x+4} - 2 \)[/tex] is obtained from [tex]\( f(x) = 2^x \)[/tex] by:
- Shifting the graph 4 units to the left,
- Reflecting the graph over the [tex]\( x \)[/tex]-axis,
- Shifting the graph 2 units downward.
Therefore, the correct description of the transformations is:
Shift 4 units left, reflect over the [tex]\( x \)[/tex]-axis, shift 2 units down.
