Let's examine the transformation from [tex]\( f(x) = 3^x \)[/tex] to [tex]\( g(x) = 3^{x+2} \)[/tex].
1. Understanding Transformation:
- The base function is [tex]\( f(x) = 3^x \)[/tex].
- The function [tex]\( g(x) = 3^{x+2} \)[/tex] can be seen as a transformation applied to [tex]\( f(x) \)[/tex].
2. Identify the Change in the Exponent:
- [tex]\( g(x) = 3^{x+2} \)[/tex] means the exponent in the base function [tex]\( f(x) \)[/tex] is altered from [tex]\( x \)[/tex] to [tex]\( x + 2 \)[/tex].
3. Interpret the Transformation:
- This transformation [tex]\( x \to x + 2 \)[/tex] inside the function represents a horizontal shift.
- When [tex]\( x \)[/tex] is replaced by [tex]\( x + c \)[/tex], the graph of the function shifts horizontally by [tex]\( c \)[/tex] units.
4. Determine the Direction of the Shift:
- If [tex]\( x \)[/tex] is replaced by [tex]\( x + c \)[/tex], the function shifts left by [tex]\( c \)[/tex] units when [tex]\( c \)[/tex] is positive.
- Here, since [tex]\( +2 \)[/tex] is added to [tex]\( x \)[/tex], the function shifts left by 2 units.
5. Conclusion:
- Thus, the correct interpretation of the transformation from [tex]\( f(x) = 3^x \)[/tex] to [tex]\( g(x) = 3^{x+2} \)[/tex] is that [tex]\( g(x) \)[/tex] moved left 2 units.
So, the transformation described is:
[tex]\[ g(x) \text{ moved left 2 units} \][/tex]
