Alright, let's break down the steps to transform the function [tex]\( f(x) = 3^x \)[/tex] into the function [tex]\( g(x) = 3^{x+1} + 4 \)[/tex].
### Steps:
1. Horizontal Shift:
- We want to change the exponent in the function from [tex]\( x \)[/tex] to [tex]\( x+1 \)[/tex].
- This means we need to shift the graph of [tex]\( f(x) = 3^x \)[/tex] horizontally to the left by 1 unit. Shifting a function [tex]\( f(x) \)[/tex] to the left by 1 unit is achieved by replacing [tex]\( x \)[/tex] with [tex]\( x + 1 \)[/tex] in the function.
2. Vertical Shift:
- Next, we need to add 4 to the entire function to shift it vertically upwards by 4 units.
- This vertical shift is achieved by simply adding 4 to the function [tex]\( f(x) \)[/tex].
### Putting it together:
- When we shift [tex]\( f(x) = 3^x \)[/tex] one unit to the left, we get [tex]\( f(x+1) = 3^{x+1} \)[/tex].
- Then, when we shift [tex]\( f(x+1) = 3^{x+1} \)[/tex] upwards by 4 units, we add 4 to the function: [tex]\( g(x) = 3^{x+1} + 4 \)[/tex].
Thus, combining both transformations, the steps to translate [tex]\( f(x) = 3^x \)[/tex] to [tex]\( g(x) = 3^{x+1} + 4 \)[/tex] are:
- Shift [tex]\( f(x) = 3^x \)[/tex] one unit to the left.
- Shift [tex]\( f(x) = 3^x \)[/tex] four units up.
Therefore, the correct answer is:
Shift [tex]\( f(x)=3^x \)[/tex] one unit to the left and four units up.
