To determine which steps will correctly translate the function [tex]\( f(x) = 3^x \)[/tex] to [tex]\( g(x) = 3^{x+1} + 4 \)[/tex], let's break down the transformations applied to [tex]\( f(x) \)[/tex] in a detailed, step-by-step manner.
First, let's examine the term [tex]\( 3^{x+1} \)[/tex]. To obtain this term from [tex]\( 3^x \)[/tex], we can rewrite [tex]\( 3^x \)[/tex] with a horizontal shift. Specifically:
- [tex]\( 3^{x+1} \)[/tex] represents [tex]\( 3^x \)[/tex] shifted one unit to the left. This is because replacing [tex]\( x \)[/tex] with [tex]\( x+1 \)[/tex] in the exponent shifts the graph one unit to the left.
Next, let's address the [tex]\( +4 \)[/tex]:
- Adding 4 to the function [tex]\( 3^{x+1} \)[/tex] implies a vertical shift of four units upward.
Putting these steps together, to transform [tex]\( f(x) = 3^x \)[/tex] into [tex]\( g(x) = 3^{x+1} + 4 \)[/tex]:
1. Shift the function [tex]\( f(x) = 3^x \)[/tex] one unit to the left to get [tex]\( 3^{x+1} \)[/tex].
2. Then shift the resulting function four units up to get [tex]\( 3^{x+1} + 4 \)[/tex].
Thus, the correct 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 and four units up.
