To transform the graph of [tex]\( f(x) = 3^x \)[/tex] to the graph of [tex]\( g(x) = 3^{x-3} + 2 \)[/tex], two transformations are needed:
1. Horizontal Translation:
- The expression [tex]\( 3^x \)[/tex] in [tex]\( f(x) \)[/tex] changes to [tex]\( 3^{x-3} \)[/tex] in [tex]\( g(x) \)[/tex]. This indicates a horizontal shift.
- Specifically, the [tex]\( x \)[/tex] value is reduced by 3, meaning the graph of [tex]\( f(x) \)[/tex] is shifted 3 units to the right.
- Answer: Horizontal translation of 3 units to the right.
2. Vertical Translation:
- In addition to the horizontal shift, we have a constant term +2 added to the function.
- This shifts the entire graph upwards by 2 units.
- Answer: Vertical translation of 2 units up.
So, to complete the statements:
- Horizontal translation of 3 units to the right
- Vertical translation of 2 units up
