Let's analyze the given functions and how they relate to each other.
1. The function [tex]\( f(x) = 3^x \)[/tex] is our original function. This is an exponential function with a base of 3.
2. The function [tex]\( g(x) = 3^x - 2 \)[/tex] is derived from [tex]\( f(x) \)[/tex]. We need to determine how this new function [tex]\( g(x) \)[/tex] compares to [tex]\( f(x) \)[/tex].
Looking closely at the equation [tex]\( g(x) = 3^x - 2 \)[/tex]:
- We see that [tex]\( g(x) \)[/tex] is formed by taking the value of [tex]\( f(x) \)[/tex] and subtracting 2.
In transformation terms, subtracting 2 from the entire function [tex]\( f(x) \)[/tex] translates the graph downward along the y-axis. Specifically, every point on the graph of [tex]\( f(x) \)[/tex] is moved 2 units lower to form the graph of [tex]\( g(x) \)[/tex].
Let's break this down step-by-step:
1. Start with the function [tex]\( f(x) = 3^x \)[/tex]. This is the original graph.
2. To form [tex]\( g(x) = 3^x - 2 \)[/tex], you take each y-value of [tex]\( f(x) \)[/tex] and decrease it by 2 units.
3. This results in a vertical shift downward by 2 units. For example:
- If [tex]\( f(1) = 3^1 = 3 \)[/tex], then [tex]\( g(1) = 3^1 - 2 = 3 - 2 = 1 \)[/tex].
- Similarly, if [tex]\( f(2) = 3^2 = 9 \)[/tex], then [tex]\( g(2) = 3^2 - 2 = 9 - 2 = 7 \)[/tex].
Every value of [tex]\( f(x) \)[/tex] is decreased by 2, indicating a vertical shift downward.
Therefore, the correct answer is:
- The graph of [tex]\( g(x) \)[/tex] is a translation of [tex]\( f(x) \)[/tex] 2 units down.
So the correct statement is:
The graph of [tex]\( g(x) = 3^x - 2 \)[/tex] is a translation of [tex]\( f(x) \)[/tex] 2 units down.
