To understand how the function [tex]\( f(x) = \log_2(x+3) + 2 \)[/tex] can be described as a transformation of the basic function [tex]\( g(x) = \log_2 x \)[/tex], we'll analyze each part of the function separately.
1. Horizontal Translation:
- The basic function [tex]\( g(x) = \log_2 x \)[/tex] becomes [tex]\( \log_2(x + 3) \)[/tex] in [tex]\( f(x) \)[/tex].
- When we replace [tex]\( x \)[/tex] with [tex]\( x + 3 \)[/tex] inside the logarithm, it indicates a horizontal shift.
- Specifically, [tex]\( \log_2(x+3) \)[/tex] means the graph of [tex]\( \log_2 x \)[/tex] is shifted 3 units to the left. This is because adding a positive constant inside the function argument [tex]\( x \)[/tex] results in a shift to the left.
2. Vertical Translation:
- The term [tex]\( +2 \)[/tex] outside the logarithm in [tex]\( f(x) = \log_2(x+3) + 2 \)[/tex] indicates a vertical shift.
- Adding 2 to the whole function [tex]\( \log_2(x+3) \)[/tex] shifts the graph up by 2 units. This is because adding a constant outside the function translates the graph vertically.
Putting these transformations together, we have:
- A translation 3 units to the left due to the [tex]\( (x+3) \)[/tex] inside the logarithm.
- A translation 2 units up due to the [tex]\( +2 \)[/tex] outside the logarithm.
So, the best description of the graph of [tex]\( f(x) = \log_2(x+3) + 2 \)[/tex] as a transformation of the graph of [tex]\( g(x) = \log_2 x \)[/tex] is:
a translation 3 units left and 2 units up.
Thus, the correct choice is:
a translation 3 units left and 2 units up.
