IDNLearn.com is the perfect place to get answers, share knowledge, and learn new things. Get the information you need from our experts, who provide reliable and detailed answers to all your questions.

Which steps will translate [tex]$f(x)=3^x$[/tex] to [tex]$g(x)=3^{x+1}+4$[/tex]?

A. Shift [tex][tex]$f(x)=3^x$[/tex][/tex] one unit up and four units to the right.
B. Shift [tex]$f(x)=3^x$[/tex] one unit up and four units to the left.
C. Shift [tex]$f(x)=3^x$[/tex] one unit to the right and four units up.
D. Shift [tex][tex]$f(x)=3^x$[/tex][/tex] one unit to the left and four units up.


Sagot :

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.