To determine the set of steps that will translate [tex]\( f(x) = 6^x \)[/tex] to [tex]\( g(x) = 6^{x-5} - 7 \)[/tex], let's break down the transformations required:
1. Horizontal Translation:
- [tex]\( f(x) = 6^x \)[/tex] transformed to [tex]\( 6^{x-5} \)[/tex]:
- When we transform [tex]\( 6^x \)[/tex] to [tex]\( 6^{x-5} \)[/tex], we are shifting the graph of [tex]\( f(x) \)[/tex] horizontally.
- Specifically, [tex]\( 6^{x-5} \)[/tex] indicates a horizontal shift of the graph five units to the right. This is because [tex]\( x-5 \)[/tex] means that we now reach the same output values five units later than we would have with [tex]\( x \)[/tex].
2. Vertical Translation:
- [tex]\( 6^{x-5} \)[/tex] transformed to [tex]\( 6^{x-5} - 7 \)[/tex]:
- When we further transform [tex]\( 6^{x-5} \)[/tex] by subtracting 7, we are shifting the graph vertically.
- Specifically, [tex]\( 6^{x-5} - 7 \)[/tex] indicates a vertical shift of the graph seven units downward.
Therefore, the correct set of steps to translate [tex]\( f(x) = 6^x \)[/tex] to [tex]\( g(x) = 6^{x-5} - 7 \)[/tex] are:
- Shift [tex]\( f(x) \)[/tex] five units to the right.
- Shift [tex]\( f(x) \)[/tex] seven units down.
So, the correct choice is:
Shift [tex]\( f(x) = 6^x \)[/tex] five units to the right and seven units down.
Thus, the correct answer is:
The second option.
