To determine the transformations of the function [tex]\( g(x) = -(2)^{x+4} - 2 \)[/tex] from the parent function [tex]\( f(x) = 2^x \)[/tex], let's break down each step involved:
1. Reflect over the x-axis:
Starting from the parent function [tex]\( f(x) = 2^x \)[/tex], to reflect the graph over the x-axis, we need to multiply the function by -1. Thus, we get:
[tex]\[
g(x) = -2^x
\][/tex]
2. Shift 4 units to the left:
Shifting a function horizontally involves adding or subtracting a value within the function's argument (inside the exponent in this case). Shifting the function [tex]\( g(x) = -2^x \)[/tex] 4 units to the left would give:
[tex]\[
g(x) = -2^{(x+4)}
\][/tex]
Adding 4 to the exponent effectively shifts the graph to the left by 4 units.
3. Shift 2 units down:
Finally, to shift the graph downward, we subtract a value from the entire function. Applying a vertical shift of 2 units down, we get:
[tex]\[
g(x) = -(2)^{x+4} - 2
\][/tex]
Combining these transformations, we see that:
- The graph is first reflected over the x-axis.
- Then, it is shifted 4 units to the left.
- Finally, it is shifted 2 units down.
Based on the step-by-step transformations:
The correct description of the transformations for [tex]\( g(x) = -(2)^{x+4} - 2 \)[/tex] from the parent function [tex]\( f(x) = 2^x \)[/tex] is:
- Shift 4 units left
- Reflect over the x-axis
- Shift 2 units down
Thus, the correct choice is:
shift 4 units left, reflect over the x-axis, shift 2 units down.
