To determine how the graph of the function [tex]\( f(x) = \frac{1}{x-2} + 1 \)[/tex] transforms into the graph of the function [tex]\( g(x) = \frac{1}{x+5} + 9 \)[/tex], we need to analyze the way these two functions are related.
First, let's inspect the parent functions inside [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]. The key parts here are the terms [tex]\( \frac{1}{x-2} \)[/tex] and [tex]\( \frac{1}{x+5} \)[/tex]. These discrepancies indicate horizontal shifts.
For [tex]\( f(x) \)[/tex]:
[tex]\[ f(x) = \frac{1}{x-2} + 1 \][/tex]
- The term [tex]\( \frac{1}{x-2} \)[/tex] indicates a horizontal shift of 2 units to the right from the basic [tex]\( \frac{1}{x} \)[/tex] function.
For [tex]\( g(x) \)[/tex]:
[tex]\[ g(x) = \frac{1}{x+5} + 9 \][/tex]
- The term [tex]\( \frac{1}{x+5} \)[/tex] indicates a horizontal shift of 5 units to the left from the basic [tex]\( \frac{1}{x} \)[/tex] function.
Now to move from [tex]\( f(x) \)[/tex] to [tex]\( g(x) \)[/tex]:
[tex]\[ \frac{1}{x-2} + 1 \rightarrow \frac{1}{x+5} + 9 \][/tex]
1. Identify the horizontal shift:
- [tex]\( \frac{1}{x-2} \rightarrow \frac{1}{x+5} \)[/tex] involves shifting 7 units to the left (because [tex]\( -2 \)[/tex] to [tex]\( +5 \)[/tex] is a shift of 7 units to the left).
2. Identify the vertical shift:
- [tex]\( 1 \rightarrow 9 \)[/tex] involves shifting 8 units up (since 9 is 8 units higher than 1).
Collectively, these shifts tell us that the transformation needed to move from the graph of [tex]\( f \)[/tex] to the graph of [tex]\( g \)[/tex] involves:
- a shift of 7 units to the left, and
- a shift of 8 units up.
Therefore, the correct statement describing the transformation is:
The graph shifts 7 units left and 8 units up.
